Investigation of channel model for weakly coupled multicore fiber

We investigate the evolution of decorrelation bandwidth of inter core crosstalk (IC-XT) in homogeneous weakly coupled multicore fibers (WC-MCFs). The modified mode-coupled equations (MCEs) are numerically solved by combining the fourth order Runge-Kutta method and compound Simpson integral method. It can be theoretically and numerically observed that the decorrelation bandwidth of IC-XT decreases with transmission distance by fractional linear function. The evolution rule of IC-XT's decorrelation bandwidth is further confirmed by experiments, which can be used as an evaluation criterion for channel model. Finally, we propose a new channel model with the coupling matrix of IC-XT generated automatically by phase transfer function (PTF), which is in good agreement with the above evaluation criterion. We believe the proposed channel model can provide a good simulation platform for homogeneous WC-MCF based communication systems.

nonlinear Schrodinger equation (NLSE) [18]. For SDM fibers, propagation effects in each spatial channel can also be described by the NLSE when there is ignorable XT between spatial channels. However, in general, the XT cannot be ignored. Therefore, to accurately evaluate the performance of SDM fiber based systems, we need to investigate the XT's effects on signal quality. Firstly, we need XT model, which is mainly focused on analyzing average XT power evolution, time-dependent characteristics, frequencydependent characteristics and other characteristics of XT caused by different fiber structures and external conditions, such as input optical power, bending, twist and other fluctuations. Further, the channel model of SDM fibers can be constructed based on XT model, which is focused on analyzing the transmission effects (especially the XT) on optical signals.
For FMF, the XT can be defined as the random power coupling between different fiber modes. The FMF's XT model shows that the coupling strength between any two modes strongly depends on the difference of propagation constant between the two modes [19]. Therefore, the modes that have similar propagation constants couple strongly. And it is always assumed that the mode coupling of different frequencies are the same [20]. The FMF's channel model has been proposed by using the coupled NLSEs based on related conclusions of FMF's XT model [21][22][23]. A differential operator can describe mode dependent loss (MDL), differential group delay (DGD) and group-velocity dispersion (GVD). And the effects of random mode coupling within a group are modeled by the generalized Manakov equations with polarization mode dispersion (PMD) taking into consideration [21,24]. In addition, the intergroup coupling is modeled using the exponent of the random Hermitian matrix [25,26]. The coupled NLSEs are always numerically solved by split-step Fourier method [18]. The existing FMF's channel model can well describe the propagation effects of optical signals in FMFs, which has been demonstrated in VPItransmissionMaker Optical Systems 9.8.
For WC-MCF's XT model, the XT can be defined as the random power coupling between different cores. The inter core crosstalk (IC-XT) can be modeled by the modified mode coupled equations (MCEs) [4,27]. Further, for homogeneous WC-MCF, when the fiber operates in the phase-matching region (when the bending radius is under the threshold bending radius), the average IC-XT power mainly changes at each phase matching points (PMPs) [4,28]. In addition, the PMPs are sensitive to the propagation constant mismatch, bending radius, twist speed and other fluctuations [28]. The average IC-XT power has been proved changing linearly with transmission distance, and bending radius with the assumption of ultra-low IC-XT [4,28]. In addition, the linear and nonlinear IC-XT power has been investigated by using the nonlinear MCEs [29]. For heterogeneous WC-MCF, the average IC-XT power evolution has been analyzed using the modified MCEs [27]. Recently, for frequency-dependent characteristics of IC-XT [30] in homogeneous WC-MCF, with the assumption of ultra-low IC-XT, the effects of group velocity's difference and GVD's difference on crosstalk transfer function (XTTF) have been discussed based on a novel analytical IC-XT model [31,32]. The model has been further improved into an analytical discrete changes model [33,34], which is suitable for analyzing the frequency-dependent IC-XT in real homogeneous WC-MCF when the fiber operates in phase-matching region. In addition, the time-dependent characteristics of frequency-dependent IC-XT have been investigated with different signal bandwidth and formats [35,36].
For WC-MCF's channel model, similar to the FMF's channel model, the linear and nonlinear propagation effects can also be modeled using coupled NLSEs with IC-XT described discretely using the coupling matrix [21,37]. However, there are two strategies to construct the IC-XT's coupling matrix for different signal bandwidths in homogeneous WC-MCF. As described in [38][39][40], for adequately narrow band signals, the IC-XT can be described as multiple delayed copies of the signals with uncorrelated phase. And for the adequately broad band signals, the IC-XT can be described as a virtual additive white Gaussian noise (AWGN) on I-Q planes due to the IC-XT changing rapidly with optical frequency [30,39]. To unify the two strategies, in our previous work [41,42], we modeled the IC-XT in frequency domain. The spectrum of optical signals is manually divided into multiple frequency segments with different IC-XT's coupling matrix. For optical signals within one frequency segment, the IC-XT is treated as signal copies with random phase according to the first strategy. Due to the IC-XT changing rapidly with optical frequency, we set the random phase of different frequency segments generated independently according to the second strategy. In [39], the IC-XT spectrum of a 17 km homogeneous WC-MCF has been measured. The decorrelation bandwidth (half width at 1/e) calculated from the autocorrelation function of the spectrum is about 10-20 pm (~1-2GHz). Therefore, we manually divide the frequency segments with the width of each frequency segment equivalent to the decorrelation bandwidth. However, there are two important issues which need be further investigated. The first issue is whether the decorrelation bandwidth (the width of each frequency segment) should be always about 1~2 GHz with the transmission distance. The second is how to construct the IC-XT's coupling matrix automatically instead of manually.
On the other hand, for SC-MCF, defined as MCF with a Gaussian-like impulse response, the core pitch of SC-MCF is always around 20-25 um [16]. The IC-XT of SC-MCF are also can be modeled using modified MCEs [16,29], which is the same as WC-MCF. When the core pitch is less than 20 um, the MCF need to be described using super-modes rather than the modified MCEs [43]. Therefore, we do not discuss the super-mode MCF in this paper. For SC-MCF's channel model, the propagation effects can also be modeled using coupled NLSEs with IC-XT described discretely using the coupling matrix due to the similarity between WC-MCF and SC-MCF. As proposed in [16], the IC-XT of SC-MCF's channel model is treated as signal copies without frequency-dependent characteristics.
Therefore, for FMF, the channel model can well model the propagation effects. But for WC-MCF and SC-MCF, the channel model need to be further discussed. For frequency-dependent characteristics of IC-XT, the channel model remains some issues to be solved as mentioned above. Although both the IC-XT in WC-MCF and SC-MCF can be modeled by the modified MCEs, the existing XT model can only suitable for analyzing the IC-XT's frequency-dependent characteristics in homogeneous WC-MCF when the fiber operates in phase-matching region. The reason is the basic expression of frequencydependent IC-XT needs the assumption of ultra-low XT and PMPs.
In this paper, to avoid the assumptions of ultra-low XT and PMPs, we choose to combine the fourth order Runge-Kutta method and compound Simpson integral method to directly solve the modified MCEs.  [4]. For homogeneous WC-MCF, the major external factors affecting IC-XT are fiber bending and twist, and the major internal factors are core pitch and the refractive index (RI) difference [28,44]. When assume the fiber is lossless, the modified MCEs are shown as Eq. (1), where the symbol A and B represent the electric fields in Core A and Core B, respectively. z is transmission distance.  is optical wavelength.    represents the coupling coefficient between Core A and Core B.   in the radial direction of the bend. We assume Core A is the center core, and Core B is the side core. When the fiber is bending and without twist, the IC-XT is ultra-low because of the mismatch of propagation constants between Core A and Core B, which is mainly caused by the inherent RI mismatch due to the actual production of fiber preform and the ERI mismatch due to the fiber bending [28]. When the fiber twists, the mismatch of the propagation constants would be zero at some points, which are called PMP. Since , eq A  and , eq B  are equal at each PMP, it will introduce a strong power coupling between Core A and Core B. For WC-MCF, we assume the electric field amplitudes of Core A and Core B are 1.0 and 0.0 at start point 0 z  . Thus, the Core A is interfering core and Core B is interfered core. With the assumption of ultra-low XT and the unchanged amplitude of Core A, the IC-XT in Core B can be represented as [4] where N B is the amplitude of Core B at Nth PMP, rnd  represents the random phase mismatch which is always assumed uniformly distributed   0, 2 . And where D is the distance between Core A and Core B.  is the twist speed. According to the central limit theorem, the real part and imaginary part of electric filed amplitude of Core B are Gaussian distributed whose variance  is for single polarization modes and 2 4  is for two polarization modes.
L is transmission distance. As shown in Eq. (6), the average IC-XT power will change linearly with bending radius and transmission length in phase-matching region.
For frequency-dependent characteristics of IC-XT in homogeneous WC-MCF, it is always assumed that the XT power is very low and the amplitude of Core A is unchanged. Therefore, the expression of IC-XT (Eq. (4)) in Core B can be rewritten as Eq. (7) when the MCF operates in phase-matching region [31][32][33].
The value K has been further investigated in [33]. The difference of propagation constants   eq   can be written in Taylor series with optical frequency  represented as Eq. (8) [18].
where 0  is the center optical frequency. 0,n  , 1, n  , and 2,n  are the Taylor series of equivalent propagation constants , eq n  of Core n .Base on the Eq. (7) and Eq. (8), the effects of group velocity walk off and GVD on frequency-dependent IC-XT has been investigated [31,32].
It should be noted that the existing XT model can only suitable for analyzing the IC-XT's frequencydependent characteristics in homogeneous WC-MCF when the fiber operates in phase-matching region.
The reason is the basic expression of frequency-dependent IC-XT needs the assumption of ultra-low XT and PMPs.

New numerical simulation method to solve the modified MCEs
To avoid the assumptions of ultra-low XT and PMPs [31][32][33], one way is to numerically solve the modified MCEs (Eq. (1)) directly. Without these assumptions, the XT in heterogeneous or homogeneous WC-or SC-MCF can be analyzed simultaneously. Since the Core A is the center core, and Core B is the side core, we rewrite the Eq. (3) with frequency-dependent characteristics taking into consideration, as shown in Eq. (9) and Eq. (10). , respectively. Bias  is the inherent propagation constant mismatch caused by fabrication error which is related to the threshold bending radius pk R demonstrated in [28]. We assume Bias  is constant with distance. And we also assume that both the core pitch D and the coupling coefficient  are constant with distance, because of the good geometric consistency of fabricated MCF. The unperturbed propagation constant c  is generated by numerically solving the dispersion equation (Eq. 3.40) in [45].  [45], which can be verified with finite element method (FEM) shown in [46]. in Eq. (4) and Eq. (7), which is always assumed uniformly distributed within  Based on the above assumptions, we can combine the fourth order Runge-Kutta method and compound Simpson integral method to solve the modified MCEs (Eq. (1)). In details, the differential equations of MCEs is solved by fourth order Runge-Kutta method [47], and the RPM (Eq. (2)) is solved by compound Simpson integral method [47]. It should be noted that the calculation step size of compound  Secondly, to verify the simulation with propagation constant mismatch, we set the bending radius 105 mm with other parameters unchanged. Fig. 2 In order to increase the calculation step size for saving time while guaranteeing the accuracy, we calculate the XT under the condition shown in Fig. 2     with core pitch 41.1 um. Therefore, the threshold bending radius is about 2.05 m [28]. Therefore, the homogeneous WC-MCF operates in phase-matching region with bending radius 105 mm. The coupling power oscillates with transmission distance and the maximum coupling power is reduced to about 7 5 10   . The reason is that the relative phase mismatch at each PMP is no longer uniformly distributed or randomly distributed when we do not add fluctuations for bending radius and twist. Fig. 4(c) shows one realization of simulation results under the third condition. We add reasonable fluctuations to make the relative phase mismatch of each PMP randomly distributed. The XT power evolution shows randomness in Fig. 4(c) rather than fixed pattern as shown in Fig. 4(b). Therefore, the maximum XT power will much larger than 7 5 10   . In addition, it can be observed that the XT power does not change linearly with transmission distance in one simulation realization.
To confirm the average XT power and XT distribution, we run the simulation for 6000 times. All the parameters are the same as those of Fig. 4(c). The fluctuations of each simulation realization are generated independently. In order to coincide with the results of Eq. (6), we carefully set Bias  to 5 2.0 10   . The reason is the average XT power is sensitive to the difference of intrinsic effective refractive index as proposed in [33]. . It can be concluded that the results of the proposed simulation method will be consistent with the existing XT model [4] with reasonable parameters. In addition, we verify the statistical distribution of XT amplitude belonging to the chi-square distribution with two-degree freedom, as shown in Fig. 5 (b). We confirm that the real part and imaginary part of XT have Gaussian profiles, as shown in In this part, we confirmed the simulation method is suitable for analyzing XT in homogeneous WC-MCF. Since the main purpose of this work is to propose the channel model for homogeneous WC-MCF in phase-matching region, we will discuss the XT with different bending radius, twist speed, propagation constant mismatch and other parameters for homogeneous ( or heterogeneous) WC-MCF ( or SC-MCF) in our future works. transmission distance has been demonstrated as Fig. 2(b) without twist and as Fig. 4(a) with twist. Fig.   6(a) show the results of frequency-dependent XT at transmission distance 400.0 m with twist (red line) or without twist (blue line). When the fiber is without twist, the optical wavelength is changing from 1549.97 nm to 1550.03 nm with 1500 sample wavelengths. When the fiber is twist, the optical wavelength is changing from 1532.00 nm to 1568.00 nm with 1500 sample wavelengths. The twist speed is 4 rounds/m. Fig.6 (b) shows the real part and imaginary part of XT without twist. It can be observed that the variation amplitudes of the real part and the imaginary part are the same. Since there is an analytical solution in the case of Fig.6 (b), we can theoretically analyze the evolution of the oscillation period with distance. We assume the refractive index is the same for different 10 a   derived by Eq. (16), which is in good agreement with fitting result 4 2. 39 10   as shown in Fig. 6(c).
In the above, we verify that the numerical simulation method can well describe the XT when there is no fluctuations and inherent propagation constant mismatch Bias  . We point out the oscillation period of frequency-dependent XT decreases with transmission distance by fractional linear function. However, for real homogeneous WC-MCF, the fluctuations and Bias  should be considered. We also calculate the XT with 1500 sample wavelengths from 1544.00 nm to 1556.00 nm under the conditions shown in    Fig. 8(b) with correlation coefficient 0.0074. Therefore, they can be treated as independent with each other. Since the XT changes randomly with optical frequency, we cannot calculate the period using Eq. (16). Therefore, we calculate the decorrelation bandwidth of XT power spectrum with half width 1 dB rather than 1/e to evaluate XT power with lower variation range. Fig. 8(c) shows the evolution of decorrelation bandwidth, which is fitted well with fractional linear function   1 az b  with 2 R higher than 0.991.
When the homogeneous WC-MCF is in phase-matching region, the XT power changes mainly at PMPs. At other positions, the XT power shows rapid oscillation with ultra-low amplitude variation.
Therefore, it is reasonable to describe that the XT happens discretely. The RPM between different cores will accumulate with transmission distance. Due to the bending, twist and other fluctuations, the RPM at each PMP is randomly distributed.
Further, the RPMs of different optical frequencies at each PMP are also different because the propagation constant will change with optical frequency. Therefore, the RPMs of different optical frequencies are different. Since the RPM at each PMP dominates the increase or decrease of XT power, the evolution of XT power of different optical frequencies at each PMP should be different, which causes the frequency-dependent characteristics of IC-XT.
In addition, the difference of RPM will increase with transmission distance, which will make the decorrelation bandwidth decrease with transmission distance. When the fiber is bending and twist and the inherent propagation constant is not zero, the RPM's differences of different optical frequencies will be approximately linearly accumulated with some fluctuations, represented as Eq. (17) based on Eq.
(2), Eq. (9) and Eq. (10). The second and third integration parts Eq. (17) will not accumulated with transmission distance because of the fluctuations without bias. Therefore, the decorrelation bandwidth of XT power decreases with fractional linear function as shown in Fig. 8 (17) 3. Experimental verification of frequency-dependent XT To verify the above conclusions, we experimentally measured the frequency-dependent XT by splicing a pair of home-made fan-in/fan-out devices at both ends of the 7-core step index homogeneous WC-MCF as shown in Fig. 9 [5].    Further, the frequency-dependent propagation constant will induce the frequency-dependent RPM during transmission. Therefore, in the coupling matrix of XT, we can assume that coupling coefficient  does not change with optical frequency. To model the propagation effects in two weakly coupled cores, the NLSEs can be represented as Eq.
In order to verify the proposed channel model, we get the evolution of XT's decorrelation bandwidth calculated by the proposed channel model. We inject an optical pulse into Core A with amplitude and initial phase 1.0 and 4  , respectively. The optical pulse is up-sampled by 8 times. The real and imaginary parts of the optical pulse are the same shown as Fig. 13(a). The spectrum of the optical pulse is shown in Fig. 13(b) with bandwidth 50 GHz. To verify the crosstalk power, we set the attenuation coefficient  to zero. To avoid nonlinear effect, the nonlinear coefficient 2 n is also zero. The center optical wavelength, the group velocity's difference and each core's dispersion are 1550 nm, 0.

Conclusions
Two issues in homogeneous WC-MCF's channel model have been investigated. For the first issue, it can be concluded that the decorrelation bandwidth of XT decreases with transmission distance by fractional linear function. In addition, The proposed numerical simulation method for solving the modified MCEs is suitable for modeling the XT in homogeneous or heterogeneous WC-or SC-MCF. For the second issue, a new channel model has been proposed based on PTF, which can describe the frequencydependent XT automatically. The group velocity difference, group velocity dispersion, and nonlinear propagation effects will affect the phase significantly, which is recorded in PTF. Therefore, the proposed channel model is suitable for analyzing the interactions between XT and the above linear and nonlinear transmission effects. The reason for the differences of decorrelation bandwidth between different cores should be further investigated. Without doubt, the channel model for homogeneous WC-MCF should be further verified by transmission experiments.