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Abstract
We numerically investigate the group delay spread (GDS) characteristics in few-mode coupled multicore fibers (FM-CMCFs) in weak random-bending conditions by using a coupled-wave theory. We qualitatively show the modal coupling between modes in the intra- and the inter-mode group generated by bending. The results indicate that the modes of FM-CMCFs are less likely to couple with each other in weak random-bending conditions in contradiction to single-mode coupled multicore fibers. To resolve this problem, we demonstrate an optimum index profile and core pitch, where the GDS can be reduced even if the different mode groups do not couple strongly.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Space division multiplexing (SDM) technology has been investigated intensively to overcome the limit of transmission capacity in a conventional single-mode single-core fiber [1]. Multicore fiber (MCF) is a type of SDM fiber and can increase the transmission channels by setting multiple spatial paths in a fiber. MCFs can be classified into non-coupled MCFs and coupled MCFs (CMCFs). In non-coupled MCFs, each guided mode is localized in an independent core and thus it requires large core pitch to avoid the crosstalk between cores. On the other hand, in CMCFs, supermodes are distributed over cores and it allows the crosstalk between cores by assumption of the use of multi-input multi-output (MIMO) processing at a receiver [2], resulting in the core pitch reduction compared to that of non-coupled MCF. In addition, CMCFs have a promising advantage that the group delay spread (GDS), which determines the system reach, can be extremely low in long-distance transmission [3] [4]. The reduction of the GDS is one of the most important issues in mode-division-multiplexed (MDM) transmission because the magnitude of the GDS determines the complexity of MIMO receiver. In CMCFs, the random inter-core modal coupling is generated by random perturbations such as random bending, twisting, splices and so on and the pulses of the supermodes that have different group delays are randomly mixed with each other. As a result, the GDS is proportional to the square root of the transmission distance [5]. (If the modal coupling is weak, the GDS is proportional to the transmission distance.) Here, we refer to this state as “strong coupling regime,” whereas we refer a state, in which the GDS is proportional to the transmission distance, as “weak coupling regime.” So far, various single-mode CMCFs (SM-CMCFs), which mean that each core supports fundamental mode only, have been reported [6] [7] [8]. To further increase the number of channels, it is desirable to expand SM-CMCFs to few-mode CMCFs (FM-CMCFs) [9] [10], in which each core supports few modes. However, there are some difficult problems in GDS reduction of FM-CMCFs. Although in Ref [9], a first experimental demonstration of strongly coupled impulse responses (IR) of 2LP-mode coupled 7-core fibers was presented, the group delays of LP01 and LP11 mode groups are significantly different (0.2 ns/km) due to insufficient coupling between two mode groups.
In this work, we investigate the GDS characteristics of 2LP-mode CMCFs in the weak bending condition by using a coupled-wave theory (CWT). First, we describe a fiber structure with a random perturbation and a GDS analysis method based on a CWT. Here, we suppose that the fibers are laid in the weak bending condition (bending radius > 1 m) and only random bending are assumed for the random perturbation. Then, we evaluate the modal coupling between modes in 2LP-mode coupled 3-core fibers (2LP-mode C3CFs) and show that the modes between different mode groups in FM-CMCFs are less likely to couple with each other. Furthermore, an optimum index profile of each core of 2LP-mode CMCFs for reducing GDS is demonstrated. For the fiber with the optimum index profile, the group velocities of two mode groups are similar and the GDS can be reduced as if all the modes are coupled strongly. Finally, the optimum core pitch for achieving the higher special density as well as small GDS in 2LP-mode C3CFs is shown.
2. GDS analysis method with random-bending model
Figures 1(a) and 1(b) show cross-sections of a coupled three-core fiber and a coupled four-core fiber, where the core radius a, and the relative refractive index difference between core and cladding is Δ. Each core is arranged in an equilateral triangle or a square with the side length of Λ, and has graded-index (GI) profile with α-parameter as shown in Fig. 1(c). In this paper, we fixed the parameters as a = 11.8 μm, Δ = 0.36%, and the wavelength λ = 1.55 μm to support 2LP modes in each core [11]. Figure 1 (d) shows the field distributions of nine supermodes of a coupled 3-core fiber. The 1st, 2nd, and 3rd supermodes are LP01-based modes (LP01 mode group), and the 4th to 9th supermodes are LP11-based modes (LP11 mode group).
Fig. 1 Cross sections of (a) coupled three-core fiber and (b) coupled four-core fiber, (c) GI profile of each core, and (d) supermode field distributions of nine modes of 2LP-mode C3CF
To consider the random modal coupling in the fiber, we employ the random-bending model [12]. Figure 2 shows the schematic of the fiber with the random-bending. In this model, the total fiber length L is divided into M segments with the segment length of ΔL. The fiber in each segment is bent with the bending radii Rx and Ry in the xz and yz plane, where x and y are transverse directions and z is longitudinal direction. The curvatures 1/Rx and 1/Ry are randomly given for each segment by Gaussian distribution with the mean value of μ [m−1] and the standard deviation of σ1/Rx, Ry = σ1/R = 1/R [m−1].
To examine the GDS of a random bending fiber, we use a CWT, in which the field amplitude and phase are fully considered, which is essential for treating the change of modal group delay due to perturbations [13]. Field coupling equations for N guided mode system in one segment are
where Am and βm are the field amplitude and propagation constant of m-th guided mode, κmn is the coupling coefficient between m-th and n-th guided modes. We suppose that the fibers are in the weak bending condition (R > 1 m), hence the supermode is applicable as the basis of CWT [14]. The coupling coefficient generated by micro-bending, κmn is assumed aswhere 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 is the normalized transverse electric field distribution of m-th mode. The refractive index distribution for bending waveguides with bending radius Rx and Ry is given byThe coupling coefficient κmn is calculated by finite element method [15]. The solution of Eq. (1) is written by using a column vector A(z) = [ A1(z) A2(z) … AN(z) ]T and a total transmission matrix T(ω) can be expressed aswhere Ti is the transmission matrix of each segment and can be obtained by diagonalizing the coupling coefficient matrix of Eq. (1) [16]. The group delay operator (GDO) is defined as [17]and the GDS is given bywhere τi is the i-th eigenvalue of GDO and is normalized as. The angled brackets denote the ensemble average. The eigenvector of GDO is so-called principal mode and the eigenvalue of GDO corresponds to the group delay of the principal mode. This quantity corresponds to the width of intensity impulse response of a multimode fiber in the strong coupling regime [18]. Here, we call this method principal mode (PM) analysis.3. Characteristics of the GDS in 2LP-mode C3CF
To estimate qualitatively the characteristics of the GDS in FM-CMCFs, firstly we investigate the modal coupling between modes in the intra- and the inter-mode group of 2LP-mode C3CFs with bending.
We define the extinction ratio as the power transferred to the other modes in the intra- or the inter-mode group when a mode is input. From Eq. (4), m-th mode extinction ratio Pm is expressed as
where tmn is the component in the m-th row and the n-th column of the transmission matrix T and G means the set of mode numbers included in each mode group.Figures 3(a)-3(d) show the core pitch dependence of the extinction ratio of each mode group as a function of the transmission distance when a LP01-based mode (1st supermode) is input, where α = ∞ (step index (SI)), μ = 1 m−1, σ1/R = 0 m−1, and ΔL = 0.01 m. These figures show the amount that the other supermodes are excited by bending when a supermode is input. As well as Fig. 3, Figs. 4(a)-4(d) show the core pitch dependence of the extinction ratio of each mode group as a function of the transmission distance when a LP11-based mode (4th supermode) is input. For any core pitch, the input mode power is transferred to the mode in the intra-mode group and the magnitude is about ~20 dB, however, the input mode power is hardly transferred to the mode in the inter-mode group because the effective index difference between mode groups is too large to couple compared to that between modes in the intra-mode group. Therefore, in FM-CMCFs, a complete random mixing hardly occurs under weak bending. Figure 5 shows the GDS after 10000-km transmission of all nine modes of the 2LP-mode C3CF with SI profile as a function of the core pitch calculated by the PM analysis, where μ = 0 m−1 and ΔL = 10 m. This propagation step size ΔL is the correlation length of polarization mode dispersion in single mode fibers [19]. The upper dashed line shows the GDS without random-bending, meaning that the GDS increases in proportional to L and the lower dashed line shows the square root of the upper line, meaning that the GDS increases in proportional to √L. The calculated results with perturbation are averaged over 30 fiber realizations. Even if σ1/R is large enough (σ1/R = 1.0 m−1), the GDS is almost identical to the upper dashed line, meaning that the fiber is in the weak coupling regime. The GDS of the fiber with perturbations is reduced for small core pitch. This is because that the coupling coefficients between different mode groups are larger than those of the fiber with large core pitch.
Fig. 3 Core pitch dependence of calculated extinction ratio of each mode group as a function of the transmission distance with Λ = (a) 25, (b) 30, (c) 35, and (d) 40 μm, when a LP01-based mode (1st supermode) is input.
Fig. 4 Core pitch dependence of calculated extinction ratio of each mode group as a function of the transmission distance with Λ = (a) 25, (b) 30, (c) 35, and (d) 40 μm, when a LP11-based mode (4th supermode) is input.
Fig. 5 GDS after 10000-km transmission of the 2LP-mode C3CF with SI profile as a function of the core pitch.
Next, we consider a GI profile because the group delays of the LP11 mode group can be controlled and the GDS between the mode groups can be reduced even if the fiber is not in the strong coupling regime. Figures 6(a) and 6(b) show the differential modal group delay (DMGD) between each supermode and 1st supermode of the 2LP-mode C3CF as a function of the α-parameter, where Λ = 30 μm. The red line shows the standard deviation of all nine supermode’s group delays. For α = ∞ (SI), the DMGDs between modes in the inter-mode group are very large. We can see that the DMGDs of the LP11 mode group are positive for α = ∞ and negative for α = 2.0 compared to those of the LP01 mode group. Hence, the DMGDs of the LP11 mode group can be greatly reduced by changing α. For α = 2.4, three modes of the LP11 mode group have the negative DMGDs and the other three modes of the LP11 mode group have the positive DMGDs. Therefore, by properly choosing α, the “averaged” DMGD of the LP11 mode group is close to 0 and the standard deviation is minimized. Figure 7 shows the GDS of all nine modes of the 2LP-mode C3CF as a function of the transmission distance calculated by the PM analysis for α = ∞ (SI), 2.4, and 2.0, where Λ = 30 μm, and ΔL = 10 m. In this paper, we set μ = 0 m−1, σ1/R = 0.6 m−1 and ΔL = 10 m, because in [20], measured and calculated results of a single-mode C3CF are in good agreement for these values. The dashed lines show the GDSs without the random-bending for each α. The calculated results with perturbation are averaged over 30 fiber realizations. For SI and α = 2, the GDSs are proportional to L and almost identical to the dashed lines, meaning that the fibers are in the weak coupling regime. On the other hand, for α = 2.4, the GDS without the perturbation (dashed line) is smaller than those of SI and α = 2, due to the “averaged” DMGD reduction shown in Fig. 6. In addition, the GDS is further reduced for the fiber with the random-bending. However, there are two unusual behaviors with respect to the results with perturbation. First, the GDS with perturbation is shows nearly √L behavior up to 10-km transmission although the modes in the inter-mode group hardly couple. Second, the √L behavior in GDS slope changed to L behavior for longer transmission. We can calculate the slopes of the GDS from Fig. 7 and they are shown in Fig. 8. The slope is normalized to be 1 when the GDS is perfectly proportional to L, thus the slope of 0.5 indicates that the GDS is proportional to √L. Interestingly, at first, the GDS slope of the fiber with α = 2.4 is low, and then increases as the transmission distance becomes longer. This is an unusual behavior because a longer transmission distance generally leads to the strong coupling regime, and then the GDS slope is reduced from 1 to 0.5 for a larger transmission distance. (The GDS of CMCFs transfers from the weak coupling regime to the strong coupling regime in a long-distance transmission.)
4. Optimum index profile for reducing the GDS
4.1 Coupled three-core fiber
In this section, we discuss the unusual GDS behavior shown in Fig. 8. To grasp the phenomenon of the increase in the GDS slope for α = 2.4 and L > 1 km, we performed the IR analysis based on multimode generalized nonlinear Schrödinger equation (GNLSE) [21,22]. The pulse propagation in FMF is described by following multimode GNLSE
where am is the complex field of mth mode, τ is the temporal coordinate normalized to LP01 mode, β0m, β1m, β2m, and αm are the propagation constant, group delay, chromatic dispersion coefficient and loss of mth mode. γmnpq and Δβmnpq are effective nonlinear coefficient and propagation constant mismatch. κmn is the random-bending induced coupling coefficient given by (2). Here, the loss and nonlinearity of the fiber are neglected since the waveform of IR of the fiber are less affected by them. Since β0m term makes the oscillation of the field very fast, it is not easy to solve the equation by split-step Fourier method (SSFM). Therefore, (8) is solved with two steps [21,22]. First, β1m, β2m terms are solved by well-known SSFM and obtain temporal field distributions including GD and chromatic dispersion effects. Next, the equation is solved for β0m and κmn terms with the temporal field obtained in the first step. In this step, analytical solutions can be used [16]. Random bending perturbations used in PM analysis is used for the perturbations inducing modal coupling in the fibre. Since the guided modes for two polarizations are degenerate, only one polarization is considered. In the following, the effect of chromatic dispersion is neglected except for the results shown in Fig. 14.Figure 9 shows IRs of the 2LP-mode C3CF with α = 2.4, where Λ = 30 μm, μ = 0 m−1, σ1/R = 0.6 m−1, and ΔL = 10 m. The calculated results are averaged over 100 fiber realizations. For these parameters, each mode group is in the strong coupling regime. Since the modes in the intra-mode group are strongly coupled and the waveforms of the same mode group are almost the same, the IRs of only 1st (LP01-based) and 4th (LP11-based) modes are shown. For 1-km transmission, the center positions of the pulses are almost the same and the pulse spread of the LP11 mode group is wider than that of the LP01 mode group. This is because that although the “averaged” DMGD of the LP11 mode group is small, the DMGDs ranges from −300 to + 300 ps/km. Thus, the GDS of the LP11 mode group is dominant. For 10- and 100-km transmission, the center positions of two pulses are separated due to the slight difference in the group delays of two mode groups and the slope of the GDS increases although each mode group itself is in the strong coupling regime. Therefore, by properly setting the “averaged” DMDG of the LP11 mode group, the √L-proportion behavior can be realized even though the modes in the inter-mode group do not couple strongly. Figure 10 shows the GDS of all nine modes of the 2LP-mode C3CF as a function of the transmission distance for α = 2.38, 2.4, where Λ = 30 μm, μ = 0 m−1, σ1/R = 0.6 m−1, and ΔL = 10 m. The dashed lines show the GDS caused by the DMGD between inter-mode group. The calculated results with perturbation are averaged over 30 fiber realizations. A slight modification to α maintains the distance of √L-proportion behavior transmission longer, and greatly reduces the GDS for the long-distance transmission. In addition, the solid lines are converged to each dashed line because the GDS is dominated by DMGD between the inter-mode group as the transmission distance become longer.
Figure 11 shows the IRs of the fiber with α = 2.38, where Λ = 30 μm, μ = 0 m−1, σ1/R = 0.6 m−1, and ΔL = 10 m. The calculated results are averaged over 100 fiber realizations. As expected, the center positions of the two mode groups are almost the same even after 1000-km transmission and the pulse of the LP01 mode group is always included in that of the LP11 mode group, meaning that the GDS is determined by the strongly coupled LP11 mode group and shows the √L-proportional behavior. From these results, there is a maximum transmission distance, at which the √L-proportion behavior transmission is possible, for each α. Figure 12 shows the GDS slope as a function of the transmission distance for various values of α. The slope for each fiber is about 0.5 for short-distance and increases to 1 as the transmission distance become longer. If we set the threshold for the slope of 0.75, the maximum transmission distances are 12 km, 50 km, and 2600 km, for α = 2.4, 2.39, and 2.38, respectively. The √L-proportion behavior transmission is possible up to 10-km in the range of α = 2.38 ~2.40. The maximum transmission distance for √L-proportion behavior can further increase by setting α-parameter more precisely, where two pulses of two mode groups are identical (or “averaged” DMGDs of the LP01 and LP11 mode groups are identical). In this work, we show that the optimum α-parameter is around 2, meaning that this value is relatively easy to fabricate, and the fabrication tolerance of α is greatly improved compared to [10].
Figure 13 shows the α-parameter dependence of the GDS after 10000-km transmission in the 2LP-mode C3CF as a function of the wavelength, where Λ = 30 μm, μ = 0 m−1, σ1/R = 0.6 m−1, and ΔL = 10 m. The dashed lines show the GDS without the random-bending for each α. From Fig. 13, for the fiber with α = 2.38, the GDS is minimum at the designed wavelength and increased for other wavelengths. The minimum GDS wavelength is shifted for the fibers with α = 2.36 and 2.40. Even if the value of α is changed by 0.02, the minimum GDS wavelength is still within C-band and considering current state-of-the-art fiber fabrication technology, α can be controlled on the order of 0.01 [23–26]. Furthermore, even if the wavelength is not the minimum GDS point, the value of GDS is one-order magnitude smaller than that of GDS calculated from DMGD. GDSs of the proposed fiber are always smaller than those of GDS calculated from DMGD for α = 2.36 to 2.40 in the wavelength range of 1530 to 1570 nm, showing the usefulness of the proposed fiber.
Fig. 13 α-parameter dependence of GDS after 10000-km transmission in the 2LP-mode C3CF as a function of the wavelength.
Finally, the effect of the chromatic dispersion is investigated. The calculated chromatic dispersion of the C3CF with Λ = 30 μm, a = 2.38 is 21 ps /nm/km. The value is consistent with that reported in [3]. Figures 14(a) and 14(b) show the waveform of 1st and 4th modes after 100 and 1000-km transmission calculated by IR analysis. The input pulse width is 40 ps. The waveform of 1st and 4th modes are almost insensitive to the chromatic dispersion. For these values, the modal dispersion is dominant for the IR waveform.
4.2 Coupled four-core fiber
To verify that the results obtained for 2LP-mode C3CFs are valid for 2LP-mode CMCFs with larger number of cores, we consider the GDS characteristics of 2LP-mode coupled 4-core fibers (2LP-mode C4CFs) with GI profile as shown in Fig. 1 (b), where Λ = 30 μm. Figures 15(a) and 15(b) show the DMGD between each supermode and 1st supermode of the 2LP-mode C4CF as a function of the α-parameter. We can see that there is an optimum index profile to achieve the “averaged” DMGD equal 0 by properly choosing α. Figures 16(a) and 16(b) show the GDS and the GDS slope of all 12 modes of the 2LP-C4CF as a function of the transmission distance for α = ∞, 2.4, and 2.38, where μ = 0 m−1, σ1/R = 0.6 m−1, and ΔL = 10 m. The calculated results are averaged over 30 fiber realizations. As in the results obtained for the 2LP-mode C3CF, the GDS can be greatly reduced for an optimum index profile since the quasi-strong coupling regime is kept. In addition, we can see that an optimum α-parameter is around 2.38 and this is identical to that of the 2LP-mode C3CF.
5. Determination of an optimum core pitch
In the previous sections, we discussed an index profile, fixing the core pitch to 30 μm. Ideally, the GDS can be more reduced as a core pitch is larger when a fiber has perfectly optimized index profile since group delays of supermode converge to that of LP01 and LP11 modes in a discrete core, which have the same group delay. However, there are always fabrication imperfections and they affect the GDS values. In addition, the strong modal coupling between modes in the intra-mode group cannot be obtained in MCFs with too small or too large core pitch [27] because it is assumed that the larger coupling coefficient and the smaller effective index difference between modes are necessary to generate the strong modal coupling in CMCFs. As the core pitch becomes small, the coupling coefficient becomes large because the mode field spreads over cores. On the other hand, as the core pitch becomes small, the effective index difference also becomes large due to the formation of supermodes [28]. Therefore, it is necessary to discuss the core pitch dependence as well as the α-parameter.
Figures 17(a) and 17(b) show the α-parameter dependence of the GDS after 10000-km transmission in the 2LP-mode C3CF as a function of the core pitch for σ1/R = 0 m−1and 0.6 m−1, respectively, where μ = 0 m−1 and ΔL = 10 m. The dashed lines show the GDS derived from the DMGD between the inter-mode group. Here, the calculated results with perturbation are averaged over 30 fiber realizations. From Fig. 17(a), we can see that the GDSs show monotonic decreasing behavior to the core pitch and converge to those derived from the DMGD between LP01 and LP11 mode for any α because the GDS between intra-mode group becomes large for small core pitch due to the formation of supermodes. From Fig. 17 (b), for α = ∞ (SI), 10, and 2, since the quasi-strong coupling do not occur at all, the GDSs hardly change even with the perturbation. On the other hand, for α ~2.38, the GDSs are fully suppressed by the perturbation even when the core pitch is small, and we can obtain almost the same GDS for all core pitch. This is because the GDSs between the intra-mode group become small due to strong coupling in each mode group and the GDSs between the inter-mode group dominate those between the intra-mode group. For α = 2.38, Λ = 29 μm, the GDS is minimized, therefore, the optimum core pitch for achieving the small GDS is around Λ = 29 μm. Finally, we compare our 2LP-mode 3CF with reported 2LP-mode 7CF. Calculated GDS of our fiber with α = 2.38 and Λ = 29 μm is 61 ps/√km and this is about 1/300 compared with that of [9] in 10000-km transmission.
6. Conclusion
The GDS of 2LP-mode CMCFs with weak random-bending are numerically investigated by using the CWT. We reveal that, although the modes in the different mode groups are difficult to couple due to the insufficient phase matching, the GDS can be greatly reduced by properly setting the “averaged” DMGD of the LP01 and LP11 mode group. We also find that there exists the maximum √L-proportion behavior transmission distance because the GDS reduction is limited by the group velocity difference between two mode groups. In addition, the optimum pitch, which gives the small GDS and the large spatial density in the long transmission distance is demonstrated. In this work, 2LP-mode C3CF structures were mainly investigated for simplicity, however, the general conclusion presented here is valid for 2LP-mode CMCFs with larger number of cores.
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