## Abstract

In this paper we evaluate experimentally and model theoretically the nonlinear crosstalk random process in multi-core fiber. The experimental results indicate that mode coupling in multi-core fibers is reduced in presence of fiber Kerr nonlinearities. An analytical study of the inter-core crosstalk probability density function in nonlinear regime is performed, where the theoretical distribution, derived from the nonlinear coupled-mode equation, is experimentally validated in homogeneous four-core fiber. The herein presented analysis includes the evaluation of the inter-core crosstalk probability density function, mean and variance evolution considering the optical power launched into the fiber.

© 2015 Optical Society of America

## Corrections

16 October 2015: A correction was made to Ref. 10.

## 1. Introduction

Spatial division multiplexing employing multi-core fiber (MCF) has been proposed to overcome the capacity limits of single-mode fiber optical systems [1]. In uncoupled MCF, inter-core crosstalk (IC-XT) is one of the most important physical impairments [2]. In recent years, linear IC-XT has been actively studied considering bending and twisting in multi-core fiber [3–7]. In linear and nonlinear regimes, the mode coupling between adjacent cores presents a stochastic nature due to random perturbations of the propagation constant of core modes induced by macrobends and structural fluctuations [7]. However, in nonlinear regime, it should be considered the nonlinear displacement of silica electrical dipoles increasing the material refractive index of each core. Therefore, similar to the behavior of the nonlinear directional coupler (NLDC) [8,9], IC-XT is reduced in nonlinear regime because the Kerr effect detunes the propagation constant of core modes [10].

This paper reports, for the first time to our knowledge, the statistical analysis of nonlinear inter-core crosstalk in good agreement with extensive experimental measurements. The paper is structured as follows: in Section 2, the nonlinear coupled-mode theory (CMT) is revisited to weakly-guiding uncoupled multi-core fibers, and the analytic probability density function (p.d.f.) of nonlinear crosstalk is derived from the nonlinear coupled-mode equation. The experimental validation of the analysis is performed in Section 3, where the p.d.f., mean and variance of the crosstalk are measured for linear and nonlinear regimes between adjacent cores showing good agreement with the nonlinear analysis herein presented. Finally, in Section 4 the main conclusions of this work are highlighted.

## 2. Coupled-mode theory and statistical analysis of nonlinear crosstalk in bent and twisted multi-core fibers

The CMT was extended to nonlinear regime by Jensen in 1982, which proposed the NLDC for all-optical switching and signal processing [8]. In the following years, the NLDC device was extensively studied. In 1989, Fraile completed the coupled mode equations of the NLDC by including additional nonlinear coupling coefficients [9]. This theory was initially developed assuming rectangular optical waveguides with transversal electric modes. In 1994, a unified approach for the coupled-mode analysis of nonlinear optical couplers based on individual linear waveguide modes was proposed by Yuan [11]. In this section, we revisited the nonlinear CMT for uncoupled MCFs including bends and structural fluctuations [7].

The nonlinear coupled-mode equation of the NLDC describing mode coupling from waveguide *m* to waveguide *n* can be expressed as [8,9,11–13]:

*A*and

_{n}*A*are the complex envelopes,

_{m}*β*and

_{n}*β*are the phase constants of bounded modes in waveguides

_{m}*n*and

*m*,

*k*is the linear coupling coefficient [14,15], and

_{nm}*q*is the self-modulation coupling coefficient [8,9,11]. The coupling coefficients are given by overlapped integrals between the eigenmodes of each core:

_{n}*ω*=

*2πc*is the angular frequency;

_{0}/λ_{0}*c*is the speed of light in vacuum;

_{0}*N*is the refractive-index distribution of the MCF [14];

^{2}*N*

_{m}^{2}is the refractive-index distribution of the MCF considering only core

*m*and cladding region [14];

**and**

*e***are the transversal vector core modes of the electric and magnetic fields, respectively;**

*h**α*is related to the third-order nonlinear susceptibility of NLDC (

*χ*

^{(3)}) as

*α*= 0.75·

*χ*

^{(3)}; and the superscript * indicates the complex conjugate term. In general, there are additional nonlinear terms and coupling coefficients in Eq. (1) of the form ∫∫

*α*|

*e*

_{m}_{(}

_{n}_{)}|

^{2}

*e*

_{m}_{(}

_{n}_{)}

*****

*e*

_{n}_{(}

_{m}_{)}

*dxdy*and ∫∫

*α*|

*e**|*

_{m}^{2}|

*e**|*

_{n}^{2}

*dxdy*[8,9,11]. Nevertheless, assuming weak mode coupling between waveguides in the NLDC, these nonlinear terms can be neglected [9]. For a waveguide-to-waveguide distance lower than the waveguide width these nonlinear terms should be included, but the CMT lacks accuracy due to the strong mode coupling among cores [9,16]. The self-modulation coupling coefficient

*q*is the strongest nonlinear term [8,9] and arises from the nonlinear interaction of a mode with itself. It is equivalent to the self-phase modulation and self-focusing effect in free space nonlinear optics.

_{n}However, it should be remarked that Eq. (1) does not apply to MCFs to describe mode coupling among cores in nonlinear regime. Macrobends and structural fluctuations of the MCF induce perturbations in the phase constants of core modes. Therefore, nonlinear CMT should be revisited for bent and twisted MCFs in coherence with [3–7,17] describing linear IC-XT. Considering a MCF with *M* cores, the equivalent phase-mismatching between core *m* and core *n* involving macrobends and structural fluctuations perturbations can be written as:

*β*and

_{n}*β*are the unperturbed phase constants of LP modes in cores

_{m}*n*and

*m*, respectively; and

*β*includes bends and structural fluctuations inducing perturbations in

_{B + S}*β*and

_{n}*β*. Therefore, the nonlinear coupled-mode equation for MCFs with the core pitch higher than the core radius can be expressed as:

_{m}

*q**= diag*(

*q*) is the

_{1},…,q_{M}*MxM*diagonal matrix of self-modulation coupling coefficients

*q*;

_{n}

*P*_{e}*= diag*(

*|A*) is the

_{1}|^{2},…,|A_{M}|^{2}*MxM*diagonal matrix including the complex envelope power;

**is the**

*β*_{eq}*MxM*diagonal matrix with the equivalent phase constants;

**is the**

*k**MxM*matrix with the coupling coefficients

*k*; and

_{nm}**(**

*E**z*) is the matrix of the electric field given by [15]:

**(**

*A**z*)

*=*[

*A*(

_{1}*z*)

*,…,A*(

_{M}*z*)]

*is the vector of the complex envelopes. The equivalent phase constants*

^{T}**depend on macrobends and structural fluctuations conditions, as depicted in Eq. (4). Slight random variations in macrobends, twist ratio and other structural fluctuations induce random changes in Δ**

*β*_{eq}*β*for every phase-matching point as reported in [17]. Therefore, linear and nonlinear inter-core crosstalk should be analyzed as a random process.

_{eq,mn}The distribution function of nonlinear IC-XT as a random process has been evaluated for the phase-matching region, i.e. where discrete dominant changes are observed at every phase-matching point, by Hayashi et al. in [17] for MCF including bends and twists. Assuming *N* phase-matching points along the MCF length, the inter-core crosstalk random variable (r.v.) from core *m* to core *n* can be defined as the ratio between the output power of core *n P _{n}*(

*z = N*) and the output power of core

*m P*(

_{m}*z = N*) when only core

*m*is excited:

*P*(

_{n}*z = N*) << 1) and short fiber lengths where the Kerr effect is stimulated, the optical attenuation can be neglected and we can approximate the output power of core

*m*as the optical power launch level,

*P*(

_{m}*z = N*) ≈

*P*(

_{m}*z = 0*) =

*P*. Thus, the nonlinear inter-core crosstalk r.v. ${X}_{nm}^{NL}$ can be written as:

_{L}*A*is the complex envelope of core

_{n}*n*. In the phase-matching region, the crosstalk longitudinal evolution from core

*m*to core

*n*can be analyzed by approximating Eq. (5) with discrete changes, in coherence with the statistical analysis of linear inter-core crosstalk depicted in [15,17]:

*N*is the

*N-th*phase-matching point,

*ϕ*is the random phase offset between core

_{mn}*m*and

*n*given by the exponential term in Eq. (5),

*L*

_{l−}_{1}is the length for the

*l−*1 longitudinal discrete change, and

*K*is the coefficient for the discrete changes caused by the coupling from core

_{nm}*m*to core

*n*[17]. In [17], Hayashi assumes that

*ϕ*is modified by slight random variations in the bending radius, twist ratio and other structural fluctuations. Considering weak coupling and low crosstalk (|

_{mn}*A*(

_{n}*z = N*)| << 1),

*A*(

_{m}*z = l−*1) can be approximated to

*A*(

_{m}*z = 0*) [17]. In addition,

*A*(

_{n}*z = l*) values can be assumed approximately similar for each phase-matching point in nonlinear regime. Therefore, considering that only core

*m*is excited, the first summation in the right-hand side (RHS) of Eq. (10) can be approximated to

*q*(

_{n}*L−L*)|

_{N}−L_{0}*A*(

_{n}*z = N*)|

^{2}

*A*(

_{n}*z = N*), where

*L*is the MCF length, and

*L*and

_{0}*L*is the length of the first and the last longitudinal discrete change, respectively. Assuming that

_{N}*L*>>

*L*, $\forall l\in \left\{0,\mathrm{...},N\right\}$, the last term can be approximated to:

_{l}*A*(

_{n}*z = N*) as:

*n*is given by the following expression:

*A*(

_{m}*z = 0*)|

^{2}is the optical power launch level

*P*, and the second term of the RHS of Eq. (14) is the linear crosstalk ${X}_{nm}^{L}$ r.v [15,17]. From Eq. (9) and Eq. (14), the nonlinear crosstalk ${X}_{nm}^{NL}$ r.v. can be written as:

_{L}*h*is a bijective and positive-real function in the domain ${X}_{nm}^{NL}\ge 0$. From [18], the p.d.f. of ${X}_{nm}^{NL}$ r.v. can be calculated as:

*u*(

*x*) is the unit step function. From Eq. (16) and Eq. (17), the probability density function of nonlinear IC-XT can be written as follow:

_{L}*g*(

*x*) is:

_{NL}Considering silica MCFs, with low nonlinear nature (*q _{n}*~10

^{−14}), Eq. (19) can be approximated to

*4x*. Therefore, in linear and nonlinear regimes, the crosstalk p.d.f. is a chi-squared distribution with 4 d.f. The mean and variance of the chi-squared distribution depend directly on the number of phase-matching points

_{NL}*N*[15], which is modified by the power launch conditions. As previously mentioned, Kerr effect mismatches the propagation constants of the LP

_{01}mode in each core. When a given core is excited with high power levels, the nonlinear displacement of silica electrical dipoles increases the material refractive index of the excited core, as given by Eq. (2,3,12) in [19]. Consequently, the equivalent refractive index difference between the excited and unexcited cores becomes larger increasing the phase-mismatching between cores and effectively reducing the crosstalk in the MCF. This behavior is similar to heterogeneous MCFs [20], where cores with different material refractive index are manufactured in order to reduce the IC-XT. Therefore, the number of phase-matching points

*N*is reduced as the power launched into the excited core increases. Since the number of phase-matching points is reduced when Kerr effect is stimulated, the mean and variance of the chi-squared distribution is also reduced in nonlinear regime.

## 3. Experimental characterization of nonlinear inter-core crosstalk distribution

In order to validate the statistical analysis in Section 2, we perform an experimental work on homogeneous four-core fiber targeting: i) To confirm the discrete random nature of nonlinear IC-XT measuring the crosstalk temporal profile, and ii) To evaluate the p.d.f., mean and variance of the crosstalk distribution considering the optical power launched into the fiber.

The crosstalk temporal profile is experimentally evaluated on 150 m of MCF with 4 cores, a constant bending radius of 67 cm and a constant twist rate of 4 turns/m. Figure 1 shows the laboratory set-up used for the evaluation of the IC-XT in linear and nonlinear regimes. A tunable external cavity laser (ECL) at 1550 nm with a linewidth of 50 kHz was used with optical amplification performed with an erbium doped fiber amplifier (EDFA) followed by a variable optical attenuator (VOA). In order to reduce the IC-XT averaging over the temporal and spectral domain due to amplified spontaneous emission (ASE) noise, the noise figure was minimized (< 4 dB) by maximizing the EDFA gain (~24 dB). The VOA was employed to modify the optical power level launched into the fiber. The optical power was injected into core 3 and measured in core 1 with a power meter (Thorlabs PM320E) every 0.5 seconds with optical power launch levels of 0 dBm in linear regime and 17 dBm in nonlinear regime.

Figure 2 shows the temporal profile of IC-XT in both power regimes during 26 hours. Discrete dominant changes can be observed in the crosstalk temporal profile in linear and nonlinear regimes as a direct consequence of the discrete longitudinal evolution of the IC-XT reported in [17] for bent and twisted MCFs. Hayashi et al. observed discrete dominant changes at every phase-matching point in linear regime between adjacent twisted cores. However, the number of phase-matching points is reduced in nonlinear regime because the fiber Kerr effect increases the equivalent refractive index difference between the excited core 3 and unexcited core 1, mismatching the effective propagation constants of core modes. Consequently, the higher the optical power level launched into core 3, the lower the crosstalk average and fluctuations measured in core 1. As the phase offsets are modified by slight random variations in the bending radius and twist ratio of the MCF due to external perturbations, the crosstalk temporal profile presents a stochastic nature in both power regimes which includes oscillatory variations arising from floor vibrations and environmental factors.

In addition, we evaluated experimentally the p.d.f., mean and variance of the inter-core crosstalk operating in linear and nonlinear regimes. The crosstalk distribution was measured between adjacent cores 3 and 1 using the same laboratory set-up depicted in Fig. 1. The experimental validation was developed using the λ-scan method [17] based on wavelength sweeping from 1550 nm to 1590 nm with 5 pm step. The optical power launch level was modified with the VOA from −3 dBm to 17.5 dBm. The ASE noise impact was evaluated measuring the crosstalk parameters with power launch conditions of −3, 0 and 3 dBm with and without the combination of EDFA + VOA. The crosstalk distribution, mean and variance were the same measuring with and without EDFA + VOA verifying that ASE noise did not average the crosstalk in our measurements.

Figure 3(a) shows the crosstalk p.d.f. for a power launch level of 0 dBm (linear regime). The measured IC-XT distribution in linear regime fits correctly to a chi-squared distribution with 4 d.f. in coherence with [15]. Considering *R _{b}* = 67 cm and

*f*= 4 turns/m conditions from the experimental set-up, and using the fiber parameters of our multi-core fiber Fibercore SM-4C1500(8.0/125)

_{T}*n*≈1.452,

_{n}*β*≈

_{n}*k*

_{0}·

*n*= 5.8851·10

_{n}^{6}rad/m and

*d*= 36 µm, we can calculate the coupling coefficients

_{nm}*k*= 0.072 m

_{nm}^{−1}and

*K*= 5.13·10

_{nm}^{−4}from [14] and [17], respectively. As can be noted, the measured distribution fits correctly to Eq. (17) with

*N*≈180. In nonlinear regime, the crosstalk p.d.f. was also identified as a chi-squared distribution, as depicted in Fig. 3(b). A chi-squared distribution with 4 d.f. was measured for a power launch level of 17 dBm. The measured nonlinear p.d.f. fits correctly to Eq. (18) with

*N*≈18. Due to the reduction of the number of phase-matching points

*N*when the optical power launch increases, the variance is also reduced in nonlinear regime, as it can be observed in Fig. 3(c). The variance decreases rapidly for power launch levels higher than 2 dBm and it stabilizes at around 15 dBm. In addition, if the number of phase-matching points

*N*is reduced with

*P*, the crosstalk mean should also decrease in presence of fiber Kerr nonlinearities. This is confirmed in Fig. 3(d) with a reduction of the IC-XT mean from −43 dB to −50 dB between both power regimes. Consequently, the IC-XT cannot be considered a strict-sense stationary random process [18] in nonlinear regime if the power launch conditions are temporally modified.

_{L}The statistical analysis performed in this section is valid for short fiber lengths where fiber Kerr nonlinearities are stimulated. After some fiber kilometers, the signal will be attenuated, Kerr effect will cease and, thus, the linear IC-XT will govern the mode coupling. For example, if the power launch level is 5 dBm and assuming a typical attenuation of 0.2 dB/km, it is expected that nonlinear IC-XT will be present for fiber lengths around 15 km. The length of nonlinear interaction is given by the following expression:

*P*is the critical power, defined as the threshold power between linear and nonlinear regimes. The critical power value depends on the nonlinear nature of silica. Jensen found an expression for the critical power of the NLDC [8]. However, in bent and twisted MCFs the solution of Eq. (5) cannot be given in terms of elliptic functions and, consequently, the critical power of the NLDC lacks validity. Nevertheless, we can analyze the critical power for MCFs considering that nonlinear optical effects occurs when the applied electric field is of the order of the characteristic atomic electric field (

_{c}*E*). For silica media, the characteristic atomic electric field is around ~10

_{at}^{5}V/m [21]. Assuming an effective area

*A*≈80 µm

_{eff}^{2}and

*E*≈1.2·10

_{at}^{5}V/m, the critical power in silica MCFs is given by the next expression [22]:

*ɛ*

_{0}and

*c*are the electrical permittivity and the speed of light at the vacuum, respectively. As can be noted, the critical power of Eq. (21) is in line with the measured results depicted in Fig. 3(d).

_{0}On the other hand, it should be remarked that for large silica MCF lengths operating with high optical power launch levels, the stimulated Brillouin scattering (SBS) will reflect a portion of the power injected. The threshold power of the SBS in silica fibers is given by [19]:

where*A*is the effective area of LP

_{eff}_{01}mode in the excited core,

*g*is the SBS gain (

_{B}*g*≈5·10

_{B}^{−11}m/W) and

*L*is the effective interaction length defined as

_{eff}*L*= [1−exp(−

_{eff}*αL*)]/

*α*, where

*α*is the fiber attenuation coefficient (0.2 dB/km ≡ 0.046 Np/km at 1550 nm in silica fibers). For large MCF lengths with

*αL*» 1,

*L*can be approximated to 1/

_{eff}*α*. It is expected that the maximum optical power injected into a single core of a MCF should be limited as in single-core single-mode fibers (~10 dBm) [23–25]. In the 150 m silica 4CF, the effective interaction length is

*L*≈0.149 km. Assuming

_{eff}*A*≈80 µm

_{eff}^{2}, the SBS threshold power level is estimated as

*P*≈23.53 dBm for the experimental setup. Consequently, for an optical power launch level of 17 dBm in 150m of 4CF, the SBS is not stimulated. SBS stimulation in short fiber lengths would require very high power levels due to the short effective interaction length of the acoustic wave generated by electrostriction.

_{th}## 4. Conclusions

In this paper, we reported the statistical analysis of inter-core crosstalk in nonlinear regime for first time to our knowledge in good agreement with experimental measurements in a homogeneous MCF compromising 4 cores. The nonlinear coupled-mode theory of the NLDC was previously extended to bent and twisted MCFs in order to perform the theoretical derivation of the nonlinear inter-core crosstalk distribution. The theoretical probability density function was also identified as a chi-squared distribution with 4 degrees of freedom for MCFs with low nonlinear nature. However, the mean and variance of the crosstalk depend on the number of phase-matching points, which are modified with the optical power launch level. Experimental verification was performed using the λ-scan method. The experimental results confirm that the self-phase modulation effect mismatches the propagation constants of the core modes reducing the number of phase-matching points in the MCF. When a given core is excited with high optical power levels, the refractive index of this core is modified by the nonlinear displacements of silica electrical dipoles. Thus, the equivalent refractive index difference between adjacent cores becomes larger increasing the phase-mismatching term. As a result, the mode coupling is reduced and the mean and variance of the chi-squared distribution decreases as the power launch level is increased. Therefore, the nonlinear IC-XT cannot be considered a strict-sense stationary random process for non-continuous wave signals. Finally, the critical power for MCF was analyzed considering the nonlinear nature of silica media. Both analytical and experimental investigations confirm that the critical power in silica single-mode MCFs is 2 dBm optical power launch.

## Acknowledgments

This work has been partly funded by Spain National Plan project MODAL TEC2012-38558-C02-01. A. Macho and M. Morant work was supported by BES-2013-062952 F.P.I. Grant and postdoc UPV PAID-10-14 program, respectively.

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