## Abstract

Inter-core crosstalk is a potential limitation on the achievable data-rates in optical fiber transmission systems using multi-core fibers. Crosstalk arises from unwanted coupling between cores of a homogenous multi-core fiber and it’s average power has been observed to vary over time by 10s of decibels, potentially requiring an additional performance margin to achieve acceptable outage probability. Most investigations of crosstalk have so far only considered continuous wave laser light or amplified spontaneous emission as sources of crosstalk. In this paper, we theoretically and experimentally investigate the time-dependence of inter-core crosstalk in a homogeneous multi-core fiber when considering signals with various modulation formats and symbol rates. We find that crosstalk power fluctuations depend on the symbol rate, modulation and skew between cores. For carrier-free signals, such as quadrature amplitude modulation, the crosstalk power is nearly constant for expected conditions of multi-core transmission systems. However, carrier-supported signals, such as OOK, always induce time-varying crosstalk powers.

© 2017 Optical Society of America

## 1. Introduction

Space-Division Multiplexing (SDM) is a very promising approach for a drastic capacity increase in short-, medium-, and long-haul fiber-optical transmission systems [1], as current single-mode fibers are reaching their capacity limits [2]. In a complete SDM system, multiple signals are transmitted through parallel propagation paths inside one optical fiber [1]. Suitable fibers for SDM are single-mode multi-core fibers, where several weakly [3, 4] or strongly coupled [5] single-mode cores are arranged inside one cladding, few- or multi-mode fibers where one single core supports several fiber modes [6] or multi-core few-mode fibers, being a hybrid fiber with several fiber cores in one cladding, each supporting more than one fiber mode [7]. While few-mode MCF may have a larger spatial information density, single-mode MCF still hold the record for the largest per-fiber capacity of 2.15 Pbps [3].

The achievable data rates in homogenous multi-core fibers can be limited by inter-core crosstalk [8], where power from a signal in one or more fiber cores couples into another fiber core, impairing a signal that travels in that core in a random noise-like fashion. Several theoretical and experimental studies have been performed to accurately characterize and model inter-core crosstalk [9–13]. It has been shown that the average crosstalk power is defined by the fiber design [10] but it has a relatively slow stochastic time dependence that can change by more than 20 dB [11]. The fluctuation speed strongly depends on the number of excited fiber cores that contribute to the overall crosstalk [14]. Similar fluctuations have been observed in the frequency domain when either sweeping an unmodulated laser [13] or modulating laser light with a single tone [11,12].

Most of the crosstalk characterizations to date have either used continuous wave (CW) signals or amplified spontaneous emission (ASE) as a source of crosstalk. Figure 1 shows the time evolution of crosstalk power, measured with the setup shown in Fig. 5, when exciting one core with either CW or ASE and observing the average crosstalk power in another core over intervals of 200 ms (short-term average crosstalk, STAXT). It can be seen that the STAXT from the ASE source hardly changes over time, while the STAXT from the CW source strongly fluctuates. This suggests that the STAXT fluctuations may be characterized as a function of the originating signals and in particular, as a function of the spectral occupation of the originating signal. This is significant for the design of MCF transmission systems and whether additional performance margins are required. As such, the aim of this paper is to investigate the impact of the choice of modulation format and symbol rate on the time-dependence of the crosstalk, and the consequences this has for the design of optical fiber systems.

## 2. Dependence of the inter-core crosstalk dynamics on the originating signal

Crosstalk in homogenous multi-core fibers occurs mainly at randomly appearing discrete points along the fiber, usually referred to as phase-matching points (PMP) [9, 10]. The total crosstalk may be approximated as the sum of the crosstalk contributions from all phase-matching points weighted by a random phase shift as well as the corresponding propagation delay, leading to a noise-like randomly distributed crosstalk signal. A schematic of this process is shown in Fig. 2.

The crosstalk can be characterized in the frequency-domain by computing the crosstalk transfer function (XTTF), given for a single quadrature of a single polarization as [11]:

*ω*is the angular frequency,

*ϕ*is uniformly distributed between 0 and 2

_{k}*π*, representing randomly occurring phase-shifts between two subsequent phase-matching points [9].

*s*is the group delay difference between core

_{nm}*n*and

*m*, referred to as inter-core skew.

*z*is the z-coordinate of the

_{k}*k*phase matching point, also being an equally distributed random variable between 0 and the fiber length

^{th}*L*. The absolute value of the discrete coupling coefficient can be written as [9]: and

*κ*,

*β*,

*R*,

*D*and

_{nm}*γ*being the mode-coupling coefficient, the propagation constant, the bending radius, the distance between cores n and m, and the twist rate respectively [9,11]. With the knowledge of the crosstalk transfer function, it is possible to calculate the crosstalk in core

*n*due to the signal in core

*m*as: We note that several works have shown that the statistical distribution of the short-term average of a narrow-linewidth

*A*(

_{n}*L*,

*ω*) can be described by a Gaussian distribution, for a sufficiently large number of PMPs [9, 11]. Furthermore, the sum of the contributions from the two quadratures and two polarizations within a core may be assumed to follow a chi-square distribution with four degrees of freedom. These statistical properties of crosstalk are valid for a CW or a single modulated tone as a crosstalk generating signal [9,11].

For simplicity, we now consider a carrier-free signal, such as QPSK or higher order QAM, in core *m* as the source of crosstalk. For simplicity, we will assume that this signal has a rectangular spectrum, such as the case of raised-cosine pulse-shaped signals. In these conditions, we may define *A _{m}*(0,

*ω*) as:

*f*is the highest modulation frequency, corresponding to the symbol rate in the case of single-carrier signals. The variance of the short term average crosstalk power in core

_{mod}*n*can be written as:

*H*(

_{XT}*L*,

*ω*). Thus, to gain some insight, we assume that the signal, and thus the crosstalk transfer function, are composed of

*M*discrete spectral lines at frequencies

*ω*that are separated by Δ

_{l}*ω*. We can then approximate Eq. (5) as:

*can than be understood as a random variable that is associated to frequency*

_{l}*ω*. In general, all Ω

_{l}*cannot be assumed independent. However, one can rewrite the variance of the sum over all Ω*

_{l}*in terms of their variances and the covariances between all Ω*

_{l}*, omitting the factor of (−*

_{l}*jK*Δ

_{nm}*ω*)

^{2}, as:

*, it is possible simplify Eq. (7) for two extreme cases of the skew and/or the modulation bandwidth.*

_{l}- If either the skew between core
*n*and*m*(*s*) is small or the signal is modulated at a low symbol rate,_{nm}*js*is much smaller than_{nm}z_{k}ω_{l}*ϕ*. Ω_{k}can then be simplified to:_{l}$${\mathrm{\Omega}}_{l}={\left|\sum _{k=1}^{N}{e}^{-j{\varphi}_{k}}{e}^{-j{s}_{nm}{z}_{k}{\omega}_{l}}\right|}^{2}\approx {\left|\sum _{k=1}^{N}{e}^{-j{\varphi}_{k}}\right|}^{2}$$From Eq. (8), it can be seen that all Ωare equal, as they only depend on the random phases_{l}*ϕ*that are the same for all frequencies_{k}*l*. Thus, the covariances Cov(Ω, Ω_{l}) in Eq. (7) are equal to the variances Var(Ω_{p}). We note that this case corresponds to having all the frequency components of the crosstalk fully correlated._{l} - When the skew between core
*n*and*m*is large and/or signals are modulated at high symbol rates, the random variable*ϕ*may be neglected. In this case, Ω_{k}can be simplified to be:_{l}

One can then write the covariance part of Eq. (7) as:

*z*are equal for all Ω

_{k}*, meaning that the arguments are strongly correlated. However, the exponential function erases these covariances, as the arguments*

_{l}*js*are multiplied with a different

_{nm}z_{k}*ω*. This corresponds to uncorrelated spectral lines, as if using ASE as crosstalk source. This effect is not trivial to show analytically so we perform extensive numerical simulations in the remainder of this section.

_{l}Figure 3 shows the covariance between all pairs of Ω* _{l}* and Ω

*for three different levels of the absolute inter-core skew, being defined as:*

_{p}*s*·

_{nm}*L*. The diagonal of all plots in Fig. 3 thus represents the variance of Ω

*. At low values of the inter-core skew (3(a)) it can be observed that the covariance is the same for all combinations of Ω*

_{l}*and Ω*

_{l}*, and thus also equal to the variances that are displayed on the center diagonal. At intermediate skew (Fig. 3(b)), the covariance is only of significant value for frequency pairs that are close by, and thus close to the diagonal of the covariance plane. For large inter-core skew (3(c)), all covariances are negligible, even those that originate from neighboring spectral components. Only the variances on the center diagonal have significant value. The red squares connect the symbol rate of the crosstalk source channel with the skew: at low skew, the symbol rate is not sufficient to reach areas where the covariance is small. However, at large skew, the entire area that is covered by the spectrum has low covariance. From Eq. (7), it can be seen that the covariances contribute*

_{p}*M*

^{2}terms and the variance only

*M*terms, showing that the covariance has a much stronger influence on the total crosstalk variance. Furthermore, it can be seen that the variance is mostly influenced by the term

*s*that represents the product of the inter-core skew and the frequency characteristics of the signal, most prominently the symbol rate that shows up in the angular frequency

_{nm}z_{k}ω_{l}*ω*. Hence, we investigate the STAXT power variance for a range of values of the skew-symbol-rate product through numerically simulations by solving the extension of Eq. (1) for dual polarization as:

**R**

*is a random unitary 2*

_{k}*x*2 matrix that accounts for random polarization rotations between phase-matching points. We only excite one polarization state at the beginning of the fiber. We consider

*K*= 10

_{nm}^{−3}and the number of phase matching points

*N*= 1000 [11]. We choose the skew-symbol-rate product, to range from 10

^{−3}to 10

^{4}, and thus a range of 7 orders of magnitude. For every value of the skew-symbol-rate product, we simulate 100000 different link realizations in order to have a sufficient number of samples for a meaningful statistic.

Figure 4 summarizes the key result of this approach and shows the variance of the crosstalk power for two different modulation formats, normalized to the average crosstalk level. At low values of the skew-symbol-rate product, the normalized crosstalk power variance reaches a value of 0.5 for both modulation formats, in accordance with published measurements with CW light as crosstalk source [9,11]. When increasing the skew-symbol-rate product, the normalized crosstalk power variation decreases, until it reaches zero for the QPSK modulated signal and about 0.12 for the OOK modulated signal. The OOK signal normalized crosstalk variance does not vanish, as the strong carrier of the signal violates the assumption that we made in Eq. (4), where we assumed the signal to have a flat spectrum. Instead, the carrier has a disproportionally stronger impact on the overall crosstalk variation compared to all other spectral components of the signal.

The transition from a normalized variance of 0.5 to 0 for QPSK and 0.12 for OOK appears roughly where the skew-symbol-rate product reaches a values of 10^{0} = 1. This is the value where the skew-symbol-rate product becomes larger than the random phase variations, and thus starts to cancel the covariances between the frequency components of the source signal. At a skew-symbol-rate product of about 10^{2} (corresponding to e.g. 10 ns total skew and a 10 GBaud symbol rate), the QPKS modulated normalized crosstalk variance approaches zero, resembling the average crosstalk from a localized component, such as a coupler or a wavelength-selective switch. This has important consequences for long-haul transmission systems, as it means in such conditions, no additional system margin is required to account for dynamic crosstalk power fluctuations. However, this is not the case for signals with a strong carrier, typical in short-haul, direct modulated transmission systems. For such systems, the average MCF crosstalk still varies stochastically, similarly to the previously observed cases with CW signals [9,11].

## 3. Experimental investigation of the crosstalk dynamics

The experimental setup for the investigation of the short term average crosstalk (STAXT) dynamics is shown in Fig. 5. Light from a distributed feedback laser (DFB), emitting at 1548.5 nm was split after a polarization controller (PC) and launched in the OOK and DP-QAM modulator paths. The LiNbO3 Mach-Zehnder OOK modulator was driven by a pulse pattern generator (PPG) that could operate with symbol rates between 0.1 and 10 GBaud. To generate a simplified example of a QAM signal with similar spectral properties, we chose single-polarization BPSK. The BPSK transmitter consisted of an optical amplifier and a LiNbO3 dual-parallel Mach-Zehnder modulator (DP-QAM), where only one quadrature of one polarization was driven by a root raised cosine filtered electrical signal, generated by a 50 GS/s arbitrary waveform generator (AWG) with symbol rates between 39 MBaud and 10 GBaud. We further generated a 24.5 GBaud DP-16QAM signal for comparison with a current state of the art modulation format, where we used a 200 kHz line-width external cavity laser instead of the DFB laser. The signals were then amplified, bandpass filtered (2 nm) and power adjusted by a variable optical attenuator (VOA). The signal was split by a 1 by 8 splitter, where six paths are sent to the outer cores of the MCF and two paths were used for power and spectral monitoring. The six excited cores of the fiber were terminated after transmission, while the STAXT was measured in the center core with a power meter, at averaging time of 200 ms, hence shorter than the timescale of STAXT changes. Each core of the fiber was designed with a step-index profile with cladding refractive index of 1.4445 and a core-cladding index difference of 0.42 % at a core pitch of 44.3 *μm*, the cladding diameter was 160 *μm*. The fiber consisted of three sections, each spooled on a reel with an average diameter of 200 mm, summing up to a total length of 53.7 km. We used laser inscribed 3-D waveguides as fan-in and fan-out devices. The crosstalk was defined as the ratio of the received power in the crosstalk-core and the through-core. Previously, it was shown [14] that the STAXT fluctuates much faster when several cores contribute to the overall crosstalk. While exciting all cores of the fiber represents a realistic transmission scenario, it moreover allows to measure better statistical properties of the crosstalk over a shorter time period. With only one core excited, the slower crosstalk dynamics could require measurements of several days, over which stable operation of all components can be difficult to ensure. Thus, in this experiment we determine the variance of the STAXT from the six outer cores to the center core after observing it for five hours per modulation format / Symbol rate. Figure 6(a) shows the normalized STAXT as a function of time with BPSK modulated signals at 39 Mbaud and 10 Gbaud for the first of the five measured hours. The STAXT fluctuates considerably at 39 MBaud, while it is almost constant at 10 Gbaud. Figure 6(b) shows the same measurement with OOK modulation at 0.1 and 10 GBaud, where both symbol rates lead to a significant STAXT fluctuation of more than 10 dB.

Figure 7 shows the variance of the normalized STAXT power as a function of the symbol rate with BPSK and OOK as well as ASE and CW as reference at normalized variance of 0 and 0.5, respectively. The variance of the BPSK STAXT vanishes at high symbol rates, while it steadily increases for smaller symbol rates. The OOK modulated signal’s STAXT has non-zero variance for all symbol rates, while in slightly increases for lower symbol rates. Neither OOK or BPSK are able to reach the upper boundary of 0.5, predicted from the theoretical analysis, since it was not possible to achieve lower symbol rates with the available components. The 24.5 Gbaud DP-16QAM signal shows zero STAXT variance. While it is beyond the scope of this paper to quantify the STAXT fluctuations for OOK signals, they clearly have a statistical behavior with nonzero variance. As such, they can potentially reach very large values, even though these are less likely and certainly upper bounded by the power of the crosstalk-generating signals.

As shown in the previous section, the STAXT variance may be calculated from the skew-symbol-rate product. Hence, for comparison we measured the skew of the investigated fiber. We transmitted a 100 ps pulse in all cores of the fiber and measured the different receiving times [15] relative to the center core. The results are shown in table 3. They neglect the small difference between the length of patch cords that connect from the power splitter to the 3-D waveguides, so that the accuracy of the measurements may only be within a few 100 ps, equivalent to about a few centimeters of single-mode fiber, being much smaller than most of the values shown in table 3.

As most absolute skews are within 1 to 20 ns, we expect the transition from low to high STAXT variance for the BPSK modulated signal to take place between 100 MBaud and 1 Gbaud, in accordance with the results shown in Fig. 7. The results emphasize that long-haul transmission systems with carrier-free modulation formats such as QPSK or higher order QAM can be designed with the assumption of a constant average crosstalk power, especially as the skew is expected to be even larger in longer systems. However, short transmission links with OOK modulation require an additional performance margin when operating in a crosstalk limited system. Due to the existence of a strong carrier component in other intensity modulated signals, such as higher order pulse-amplitude modulation (PAM), similar behavior must be expected. This needs to be taken into account e.g. when designing fibers with high core-count to maximize the spatial information density that might be useful in intra- and inter-data center communication. Hence, this result has important consequences for the design of optical transmission systems using multi-core fiber and answers some of the questions about their use as practical transmission fibers.

## 4. Conclusion

We have analyzed the dependence of the short term average crosstalk power variations on the inter-core skew and the modulation format / symbol rate of the transmitted signals in homogeneous single-mode multi-core fibers through analytical, numerical and experimental investigations. The results indicate that the power fluctuations are mainly induced by a correlation between different spectral components that can be relieved by either transmitting through fibers with large inter-core skew or using moderate to high symbol rates. However, signals that contain a strong carrier, like OOK or PAM4, maintain a level of crosstalk power variation, even for large skew and symbol rates. Thus, intensity modulated systems likely require a performance margin even at high symbol rates and skew values. In contrast, the observation of low crosstalk variation for higher order modulation formats suggests that no such margin is required for systems using these formats.

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