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The effect of bending and twisting on the group delay spread (GDS) of strongly coupled 3-core fibers is investigated. For the random perturbation inducing modal coupling in the fiber, two physical mechanisms, microbending or macrobending with random twist, are considered. Calculated results show that both mechanisms lead to the same effect, namely, reduced GDS under strong coupling regime. Furthermore, a novel fiber structure having an air-hole at the center is proposed for reducing the GDS. By placing the air-hole, the effective index difference between fundamental and the higher order modes is reduced, leading to strong modal mixing in the fiber, and hence, small GDS. Calculated GDS of the fiber with air-hole is almost 1/5 compared with that of the fiber without air-hole.
© 2016 Optical Society of America
1. Introduction
Mode-division-multiplexing (MDM) transmission technique has attracted considerable attention as a promising technology for increasing network capacity. Few-mode fibers (FMFs) used for MDM transmission have also been intensively studied and various designs of FMF have been proposed [1]. One of the major problems in FMF transmission is the group delay spread (GDS) originating from the modal dispersion since the magnitude of GDS determines the complexity of MIMO receiver [2] (and hence, system reach). GDS is proportional to the transmission distance if the coupling between different fiber modes is weak (weak coupling regime). If the modal coupling is strong (strong coupling regime), the GDS is reduced, and it is proportional to the square root of the transmission distance [3]. Recently, this square root dependence of GDS was experimentally shown by using well-designed strongly-coupled 3-core fibers (3CFs) up to 6000 km [4]. The behaviour was theoretically analyzed by using Stokes space analysis and numerical Monte-Carlo simulation [5]. In [5], a core-diameter deformation as well as core-to-core distance fluctuations and intra-core random birefringence were considered for the perturbation inducing mode-coupling effect. Recently, it was experimentally pointed out that a macrobending and fiber twisting, which is frequently used model for the crosstalk analysis of multicore fibers (MCFs), can be a dominant mechanism for random modal coupling [6–8].
In this paper, the effect of bending and twisting on the GDS of strongly coupled 3CFs is investigated. For the random perturbations inducing modal coupling, we consider two mechanisms: microbending [9,10] and macrobending with twist. It is shown that from the theoretical point of view, these mechanisms give the same effects for the GDS, namely, reduced GDS under strong coupling regime. Calculated GDS in strong coupling regime shows square root dependence in terms of transmission distance. Furthermore, we propose a new fiber structure for reducing the GDS. It is shown that the GDS can be reduced further by adding an air-hole to the centre of the fiber, leading to additional freedom of fiber design for minimizing GDS.
2. Fiber structure and random perturbations
Figure 1(a) shows the cross-sectional structure of the fiber considered here. Three identical cores are arranged in an equilateral triangle with the side length of Λ. The core radius is a = 6.2 μm and the refractive index difference, Δ = 0.27% [4] (normalized frequency, V = 2.22). The wavelength is 1.55 μm. An air-hole is added to the center of the fiber and the radius is a2. Figure 1(b) shows the electric field distributions of three supermodes of this fiber without air-hole for Λ = 29 μm [4]. First mode has almost uniform distributions for three cores. Second and third modes are degenerate and have zero intensity at the center of the fiber. For this fiber, calculated differential modal group delay (DMGD) between first and second (third) modes without random modal coupling is 212 ps/km. The modes are degenerate in terms of polarization.
Fig. 1 (a) The cross section of the fiber structure and (b) guided mode field distributions of 3CF without air hole.
For the perturbations inducing modal coupling, microbending or macrobending with twist is taken into account. Figure 2(a) shows the schematic of the fiber with microbending. The total fiber length (L) is divided by M segments with constant segment length ΔL as shown in Fig. 2(a) and random bending curvatures, Rx and Ry, are given for each section. The distribution of the inverse of the curvatures is Gaussian distribution with the mean value of 0 m−1 and the standard deviation (STD) of σ1/Rx = σ1/Ry = σ1/R m−1. For σ1/R = 0 m−1, the section is straight. Here, x and y are transverse and z is longitudinal directions. To take into account polarization mixing effect, the fiber twist is incorporated between each segment. The twisting angle between each segment Δθi is randomly generated by Gaussian distribution with the mean value of 0 rad and the STD of σθ rad.
Figure 2(b) shows the schematic of the fiber with macrobending and twist [11]. The fiber is bent to x direction with the constant bending radius of R. It should be noted that in the experiment, since the fiber is usually coiled around the fiber spool, this is more realistic situation. The fiber is divided into M segments as in Fig. 2(a), and each segment has randomly generated twist angle Δθi relative to the previous segment. Δθi has Gaussian distribution with the mean value of 0 rad and the STD of σθ rad. Therefore, twisting angle of ith segment is θi-1 + Δθi. Figure 2(b) shows the example of fiber cross section of θ = 0, π/4, and π/2. For both models, it is assumed that the structure is uniform within one segment.
3. Coupled wave theory and the group delay spread
To treat the GDS, we use coupled wave theory, in which the field amplitude and phase are fully taken into account, which is essential for treating the change of modal group delay due to perturbations [12]. Field coupling equations for N guided mode system in one segment are
where am and βm are the field amplitude and propagation constant of mth guided mode including polarizations. The order of the modes are a1 is E1x, a2 is E2x, a3 is E3x, a4 is E1y, a5 is E2y, and a6 is E3y. Here, Emx and Emy stand for x- and y-polarization of mth mode. κmn is the coupling coefficient between mth and nth guided modes. For the microbending, it is given bywhere ω is the angular frequency, ε0 is the free-space permittivity, nbend and nst are the refractive index distributions for bending and straight waveguides, Em and Hm are transverse electric and magnetic field distributions of mth mode. For the macrobending with twist model, κmn is given byHere, x’ = xcosθ−ysinθ and y’ = xsinθ + ycosθ are the rotated coordinate with respect to the original ones with the rotation angle θ. These expressions for coupling coefficients can be easily derived [13] by assuming that Em and Hm satisfy Maxwell’s equations for unperturbed waveguides (in this case, straight waveguide without any bending and twisting) and the electromagnetic fields in perturbed waveguides can be expressed by the sum of these eigenmodes. Here, the amplitudes of Em and Hm are related to the optical power P asRefractive index distribution under the bending radius R is given by
In Eq. (3), Em and Hm are also rotated field distribution. Finite-element method [14] is used for the calculation of the coupling coefficient. In this model, κmn = κnm. Figure 3 shows κmn for x-polarization of macrobending with twist model for R = 5000 mm as a function of θ(/π). κmn is dynamically changed with θ. Since this model assumes the straight waveguide as unperturbed system, the model cannot be used for strongly bent condition as in [6]. We use R = 5000 mm hereafter, since the bending loss of guided modes is on the order of 10−10 dB/km and can be neglected.The solution of (1) can be written as
where Ti is the transmission matrix of each section. Ti is given by two matrix products, namely, Ti = RiUi, where U is the matrix for the mode coupling given by (1). We assume that U is block diagonal in terms of polarization [12], and therefore, polarization mixing does not occur in the segment. Polarization mixing is considered by R, which is the rotation matrix between two segments originating from the twist. The rotation matrix is simply given by [12]By using total transmission matrix, T, we can define so-called group delay operator (GDO) as [15]
The GDS, σgd, is given by [16,17]where τi is the ith eigenvalue of GDO and is normalized as Στi = 0. The angled brackets denote the ensemble average. It was recently revealed in [16] that this quantity corresponds to the width of intensity impulse response of the recent MDM fiber [4] in the strong coupling regime. ΔL is taken as 10 m otherwise noted, which is the typical polarization mode dispersion correlation length of single mode fibers [5].4. Group delay spread of strongly coupled 3CF
4.1 3CF without air hole
Here, we consider 3CF without air hole. The fiber parameters are from [4], (a = 6.2 μm, Δ = 0.27%, and Λ = 29 μm). Figure 4 shows σgd as a function of transmission distance using microbending model. σθ = 0.6 rad [12]. Upper dashed line shows the σgd without modal coupling and lower dashed line shows the square root of upper line. Solid lines are calculated σgd for various values of σ1/R. The results are obtained by averaging over 300 realizations. For σ1/R = 0.001 m−1, σgd increases linearly with L and for large values of L, it deviates from linear increase (weak coupling regime). For σ1/R = 0.01 m−1, up to 10-km, σgd is proportional to L. For L larger than 100-km, the increase of σgd is suppressed due to the mode coupling and is proportional to L0.5 (strong coupling regime). For larger values of σ1/R, σgd shows square root dependence for L, which is consistent with the measured results, and for σ1/R = 0.6 m−1, calculated results are well fitted to reported measured results [4]. It should be noted that although only microbending effect is taken into account for this model, one can consider that other perturbations, such as strain-optical effect [3] and core deformations [5], are included in σ1/R.
Fig. 4 σgd as a function of transmission distance of 3CF without air hole calculated by microbending model (R = 5000 mm).
Figure 5 shows σgd as a function of transmission distance using macrobending with twist model. Qualitatively, the same results as in Fig. 4(a) is obtained. For this model, theoretical results are well fitted to the reported measured values [4] for σθ = 2 rad. From these results, both microbending and macrobending with twist models give the same results for GDS. Both mechanisms randomize the transmission matrix T and reduce the variance of GD [3]. Hereafter, we use macrobending with twist model. It should be noted that since the guided modes are degenerate in terms of polarization, the results obtained by both models are almost the same even if the polarization mixing effect is neglected.
Fig. 5 σgd as a function of transmission distance of 3CF without air hole calculated by macrobending with twist model (R = 5000 mm).
In [6], it was reported that there is a possibility that ΔL is much smaller than 10 m. Therefore, ΔL dependence of σgd is investigated. Figure 6 shows σgd as a function of the correlation length, ΔL, for the transmission distance of 100 km. For the same σθ, σgd is decreased for smaller ΔL. This is because if ΔL is small, the transmission matrix is more randomized, leading to smaller σgd. For σθ = 0.01 rad, σgd is not so changed for ΔL between 10 and 100 m. Since for this transmission length and σθ, the fiber is in weak coupling regime, σgd is mainly determined by DMGD.
4.2 3CF with air hole
Next, we consider 3CF with air hole at the center of the fiber to reduce σgd further. Figure 7 shows the electric field distributions of three supermodes of the fiber with air hole, for Λ = 29 μm and a2 = 6.2 μm. For first mode, due to the air hole, the field at the center of the fiber is smaller than that of fiber without air hole (Fig. 1(b)). For second and third modes, since the amplitudes at the fiber center are zero for the fiber without air hole, the field distributions are less affected by air hole than the first mode. Therefore, effective refractive index, neff, of the first mode is reduced much, while neff of the higher order modes is not so changed, leading to smaller neff difference (Δneff) and strong mode coupling. Figure 8 shows DMGD and Δneff as a function of Λ for various values of a2 calculated by finite element method. For Δneff, the value is reduced for larger values of a2 as expected. DMGD is also reduced for larger values of a2. Therefore, the modal coupling is stronger for the fiber with air hole for the same Λ. For the same Δneff, we can reduce Λ, and can increase the space utilization efficiency [18]. Figure 9 shows κmn for x-polarization of 3CF with air-hole for R = 5000 mm and a2 = 6.2 μm as a function of θ(/π). κmn is dynamically changed with θ and there are no noticeable difference with Fig. 3. Figure 10 shows σgd as a function of transmission distance of 3CF with air hole (a2 = 6.2 μm). The macrobending radius is R = 5000 mm. Compared with the results of Fig. 4 (b), σgd is smaller for the same value of σθ, since the DMGD and Δneff are smaller than that of the fiber without air hole. σgd of the fiber with air-hole can be reduced almost 80%, leading to the simpler configurations of MIMO receiver.
Fig. 9 κmn of macrobending with twist model as a function of θ(/π) of 3CF with air-hole (a2 = 6.2 μm, R = 5000 mm).
5. Conclusion
We have investigated the effect of bending and twisting on the modal GDS of 3CFs. For the perturbation, we consider microbending or macrobending with twist model and it is shown that both models have the same effect on GDS. Furthermore, it is demonstrated that by placing an air hole at the center of the fiber, σgd can be further reduced, leading to simpler configuration of MIMO receiver. These results add new degree of freedom for the strongly-coupled FMF design for MDM transmission.
Acknowledgment
The authors acknowledge the suggestions from Dr. A. Meccozi. A part of this work was supported by the National Institute of Information and Communication Technology (NICT), Japan under “R&D of innovative Optical Fiber and Communication Technology”.
References and links
1. P. Sillard, M. Bigot-Astruc, and D. Molin, “Few-mode fibers for mode-division-multiplexed systems,” J. Lightwave Technol. 32(2), 2824–2829 (2014). [CrossRef]
2. S. O. Arik, J. M. Kahn, and K.-P. Ho, “MIMO signal processing for mode-division multiplexing,” IEEE Signal Process. Mag. 31(2), 25–34 (2014). [CrossRef]
3. K.-P. Ho and J. M. Kahn, “Statistics of group delays in multimode fiber with strong mode coupling,” J. Lightwave Technol. 29(21), 3119–3128 (2011). [CrossRef]
4. R. Ryf, R.-J. Essiambre, A. H. Gnauck, S. Randel, M. A. Mestre, C. Schmidt, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, T. Hayashi, T. Taru, and T. Sasaki, “Space-division multiplexed transmission over 4200-km 3-core microstructured fiber,” in Proc. OFC (2011), paper PDP5C2.
5. C. Antonelli, A. Mecozzi, and M. Shtaif, “The delay spread in fibers for SDM transmission: dependence on fiber parameters and perturbations,” Opt. Express 23(3), 2196–2202 (2015). [CrossRef] [PubMed]
6. T. Hayashi, R. Ryf, N. K. Fontaine, C. Xia, S. Randel, R.-J. Essiambre, P. J. Winzer, and T. Sasaki, “Coupled-core multi-core fibers: High-spatial-density optical transmission fibers with low differential modal properties,” in Proc. of ECOC (2015), paper We.1.4.1. [CrossRef]
7. T. Sakamoto, T. Mori, M. Wada, T. Yamamoto, and F. Yamamoto, “Fiber twisting and bending induced mode conversion in coupled multi-core fiber,” in Proc. ECOC (2015), paper P.1.2.
8. T. Sakamoto, T. Mori, M. Wada, T. Yamamoto, F. Yamamoto, and K. Nakajima, “Fiber twisting and bending induced adiabatic/nonadiabatic super-mode transition in coupled multi-core fiber,” J. Lightwave Technol. 34(4), 1228–1237 (2016). [CrossRef]
9. A. A. Juarez, E. Krune, S. Warm, C. A. Bunge, and K. Petermann, “Modeling of mode coupling in multimode fibers with respect to bandwidth and loss,” J. Lightwave Technol. 32(8), 1549–1558 (2014). [CrossRef]
10. T. Fujisawa and K. Saitoh, “A principal mode analysis of strongly-coupled 3-core fibers,” in Proc. ECOC (2015), paper We.1.4.6.
11. K. Saitoh, “Multicore fiber technology,” in Proc. OFC (2015), paper Th4C.1.
12. M. B. Shemirani, W. Mao, R. A. Panicker, and J. M. Kahn, “Principal modes in graded-index multimode fiber in presence of spatial- and polarization-mode coupling,” J. Lightwave Technol. 27(10), 1248–1261 (2009). [CrossRef]
13. K. Okamoto, Fundamentals of Optical Waveguides (Academic Press, 2006).
14. T. Fujisawa and M. Koshiba, “Finite element characterization of chromatic dispersion in nonlinear holey fibers,” Opt. Express 11(13), 1481–1489 (2003). [CrossRef] [PubMed]
15. S. Fan and J. M. Kahn, “Principal modes in multimode waveguides,” Opt. Lett. 30(2), 135–137 (2005). [CrossRef] [PubMed]
16. A. Mecozzi, C. Antonelli, and M. Shtaif, “Intensity impulse response of SDM links,” Opt. Express 23(5), 5738–5743 (2015). [CrossRef] [PubMed]
17. C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express 20(11), 11718–11733 (2012). [CrossRef] [PubMed]
18. T. Sakamoto, T. Mori, M. Wada, T. Yamamoto, and F. Yamamoto, “Coupled multicore fiber design with low intercore differential mode delay for high-density space division multiplexing,” J. Lightwave Technol. 33(6), 1175–1181 (2015). [CrossRef]
