Birefringence effects in multi-core fiber : coupled local-mode theory

In this paper, we evaluate experimentally and model theoretically the intraand inter-core crosstalk between the polarized core modes in single-mode multi-core fiber media including temporal and longitudinal birefringent effects. Specifically, extensive experimental results on a four-core fiber indicate that the temporal fluctuation of fiber birefringence modifies the intraand inter-core crosstalk behavior in both linear and nonlinear optical power regimes. To gain theoretical insight into the experimental results, we introduce an accurate multi-core fiber model based on local modes and perturbation theory, which is derived from the Maxwell equations including both longitudinal and temporal birefringent effects. Numerical calculations based on the developed theory are found to be in good agreement with the experimental data. © 2016 Optical Society of America OCIS Codes: (060.0060) Fiber optics and optical communications; (060.2270) Fiber characterization; (060.2300) Fiber measurements; (060.4370) Nonlinear optics, fibers. References and links 1. B. J. Puttnam, R. S. Luis, W. Klaus, J. Sakaguchi, J.-M. Delgado Mendinueta, Y. Awaji, N. Wada, Y. Tamura, T. Hayashi, M. Hirano, and J. Marciante, “2.15 Pb/s transmission using a 22 core homogeneous single-mode multi-core fiber and wideband optical comb,” in Eur. Conf. Opt. Commun. (ECOC) (2015), paper PDP 3.1. 2. T. Mizuno, H. Takara, A. Sano, and Y. Miyamoto, “Dense space-division multiplexing transmission systems using multi-core and multi-mode fiber,” J. Lightwave Technol. 34(2), 582−591(2016). 3. M. Morant, A. Macho, and R. Llorente, “On the suitability of multicore fiber for LTE-advanced MIMO optical fronthaul systems,” J. 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Introduction
Space-division multiplexing (SDM) using single-mode multi-core fibers (SM-MCFs) has attracted much attention over the last years to provide large fiber transmission capacity in backbone and access optical networks [1][2][3].However, advanced SDM transmissions in polarization multiplexing SM-MCF systems should deal with the intra-and inter-core crosstalk impairment, which presents both a spatial and temporal stochastic nature induced by different effects.The spatial dependence of the crosstalk is induced by the longitudinal fiber perturbations as macrobending, microbending, and fiber twisting [4].Moreover, the temporal dependence of the crosstalk is induced by: the relative delay between cores [5], the walk-off effect due to different group velocities between cores and referred to as the inter-core skew [6], changes in the state of polarization of the input signal [7] or the MCF temporal birefringence fluctuations.
In this topic, the spatial random nature of the crosstalk has been extensively investigated in [8][9][10][11][12][13] considering a single polarization and neglecting the intra-core coupling between orthogonal polarizations induced by the longitudinal random perturbations, which were heuristically inserted in the coupled-mode equations.On the other hand, in [14][15][16][17] the Manakov equations have been extended to multi-mode fibers (MMFs) and multi-mode multi-core fibers (MM-MCFs) to analyze linear and nonlinear propagation in SDM systems modeling intra-and inter-core coupling effects.In addition, in [18,19] the mode coupling has also been theoretically investigated in MMFs and MM-MCFs considering both orthogonal polarizations and the longitudinal random perturbations of the fiber.In [18] the bending and twisting of the media were modeled in the coupling matrix (composed by the coupling coefficients of the coupled-mode theory proposed in [20] for anisotropic optical waveguides), and in [19] the spatial random perturbations were included in the propagation matrix of the optical media.In all these previous works [8][9][10][11][12][13][14][15][16][17][18][19], the longitudinal perturbations of the fiber have been described assuming ideal modes but omitting the temporal fluctuations of the SDM fiber perturbations.
Furthermore, the temporal dependence of the crosstalk in SM-MCFs due to the relative delay between cores, the inter-core skew and changes in the state of polarization of the input signal has been investigated in [5][6][7] including the fiber perturbations in the coupled-mode theory using a heuristic formalism and omitting intra-core and nonlinear effects.In addition, the temporal fiber birefringence fluctuation inducing crosstalk temporal dependence has not been investigated so far in SM-MCFs.
This paper reports on the theoretical and experimental investigation of the temporal MCF birefringence effects and longitudinal random perturbations and their impact on the intra-and inter-core crosstalk in SM-MCFs using a more rigorous formalism.In particular, aimed to accurately describe linear and nonlinear intra-and inter-core coupling effects in SM-MCFs including both longitudinal and temporal MCF perturbations, we propose and develop a coupled local-mode theory (CLMT) considering these random perturbations in the Maxwell equations from the onset.As a result, we obtain an accurate fiber model able to predict the different types of crosstalk between the polarized core modes (PCMs) in SM-MCFs: (i) the intra-core crosstalk (iC-XT) which describes the mode coupling between orthogonal polarizations in a given core; (ii) the direct inter-core crosstalk (DIC-XT) modeling the mode coupling between the same polarization axis in different cores; and (iii) the cross inter-core crosstalk (XIC-XT) involving mode coupling between orthogonal polarizations in different cores.To complete our study, we perform extensive experimental measurements on a four-core fiber, which are found to be in good agreement with numerical simulations based on the proposed CLMT theory.Considering that in previous works [7][8][9][10][11][12][13] the crosstalk in MCF is usually estimated calculating the mean of the random process, our investigations are mainly focused on the behavior of the crosstalk mean between the PCMs.
The paper is structured as follows.In Section 2, the aforementioned coupled local-mode theory is derived from the Maxwell equations.In Section 3, the fiber simulation model is described and extensive numerical simulations are performed considering lowly-and highlybirefringent (LB and HB) cores.The temporal evolution of the MCF random birefringence and its impact on the mean of the linear and nonlinear iC-, DIC-and XIC-XT is also reported.The experimental validation of the theoretical model is performed in Section 4 on a homogeneous four-core fiber (4CF).Finally, in Section 5 the main conclusions of this work are highlighted.

Coupled local-mode theory
Coupled local-mode theory (CLMT) for SM-MCF media including both temporal and longitudinal birefringent effects in the Maxwell equations is reported in this section.As depicted in Fig. 1, in a real two-core MCF, each core m = a,b can be modeled as a series of birefringent segments with a different time-varying retardation and random orientation of the local principal axes.Therefore, the first-order electrical susceptibility tensor χij (1) can slightly fluctuate along the fiber, as a consequence of external perturbations such as bending, twisting, and temperature variations, as well as due to manufacturing imperfections [4,21].As a result, in each segment of a given core m, the propagation constant of the PCMs LP01x and LP01y presents a different value due to the mentioned slight changes, and therefore, the transversal function of each PCM "mi" (i = x,y) is also modified.
In order to model theoretically this scenario, the concept of local mode is included in the classical perturbation theory [22].A local mode can be defined as an eigenfunction in a short core segment where the propagation constant and the transversal Bessel function [23] are approximately invariant.Hence, each core can be separated in different segments and In this way, in contrast with [4][5][6][7][8][9][10][11][12][13] where the longitudinal and temporal MCF perturbations are heuristically inserted in the coupled-mode theory assuming ideal modes, we include the longitudinal and temporal MCF random perturbations in the Maxwell equations considering local modes.The resulting coupled local-mode equations include oscillatory terms and mode-coupling coefficients (MCCs), which are longitudinal and temporal dependent.This CLMT formalism will allow us to model the impact of the different longitudinal and temporal fiber random perturbations on the crosstalk among the different PCMs and find the predominant linear and nonlinear MCCs describing iC-XT, DIC-XT and XIC-XT.

Nonlinear coupled wave equations
Our initial goal is to include the MCF random perturbations in the theoretical analysis from the Maxwell equations.We start by analyzing the coupled wave equations of an anisotropic and nonlinear SM-MCF.Considering monochromatic electric fields with both orthogonal polarizations and slowly varying amplitude functions, the real wave function of the global electric field (Ɛ), the linear polarization (Ƥ (1) ) and the nonlinear polarization (Ƥ (3) ) of a weakly guiding MCF can be written as [24][25][26]: where Ei,ω0, Pi,ω0 (1) and Pi,ω0 (3) are the complex amplitudes, ω0 is the angular frequency of the optical carrier and c.c. are the complex conjugate terms.In this work, the semicolon symbol in Eqs. ( 1) is used to separate the rapid temporal oscillation of the optical carrier from the slow temporal evolution of the complex amplitudes modeling temporal birefringence fluctuations or additional temporal variations as the fluctuation of the state of polarization of the input signal.Although the MCF dispersive effects inducing crosstalk time dependence [27] have been omitted in Eqs.(1) when assuming monochromatic electric fields, they will be discussed at the end of section 2.3.In addition, the complex amplitude vector Pi,3ω0 (3) of the nonlinear polarization in 3ω0 was omitted, taking into account that the phase-matching condition in this nonlinear term is not satisfied in silica fibers [24].The complex amplitude of the linear and nonlinear polarizations can be expressed as [25]: where the first-order electrical susceptibility tensor χij (1)  is assumed longitudinal and temporal dependent due to the MCF perturbations.Moreover, the third-order susceptibility χijkl (3)  can be approximated in silica media to [26]: where χxxxx (3)  = χijkl (3)  with i = j = k = l = x and δij is the Kronecker delta function defined as δij = 1 when i = j and δij = 0 otherwise.
From the first and second Maxwell equations (Ampère's and Faraday's laws) in the time domain and using Eqs.( 1)-( 3), the nonlinear coupled wave equations for the complex amplitudes are found to be: ; ; ; ; 0, 3 ; ; ; ; 0, 3 where Δ is the Laplacian operator; εr,i = 1+χii (1) ; σ = χxy (1)  ; γ = 0.75•χxxxx (3) ; k0 = ω0/c0 is the wavenumber at the vacuum and Ei,−ω0 = (Ei,ω0) * , where * denotes complex conjugation.Note that we have assumed in Eq. (4b) that χxy (1)  = χyx (1)  in order to satisfy the Poynting theorem [28] when considering monochromatic electric fields in silica media.Now, in order to simplify the mathematical discussion and without loss of generality, we consider a MCF comprising two cores a and b.Thus, the relative electric permittivity of the MCF εr,i is defined in each polarization axis i = x,y as (we use the symbol := to indicate definition): where εr,ai, εr,bi and εr,ci are the relative electric permittivity in the cores a, b and in the cladding, respectively; Δεr,ai and Δεr,bi are the difference between the relative electric permittivity of the cladding and those of cores a and b, respectively; and nai, nbi and nci are the refractive index in each polarization axis in cores a, b and in the cladding region, respectively, considering both LB-and HB-MCFs.The linear birefringence is observed when εr,x ≠ εr,y and the circular birefringence is given by: Furthermore, considering the low nonlinear nature of silica media, the nonlinear parameter γ is assumed constant in each dielectric region of the MCF.

Coupled local-mode equations
From the perturbation theory [22], the electric field of Eqs.(4) can be approximated in the MCF media as (i = x,y): ; ; , , ; , exp where Emi is the complex amplitude of the electric field of the PCM "mi" considering isolated cores; Ami is the complex envelope; Fmi is the transversal local eigenfunction which can be calculated using the equivalent refractive index nmi (eq) of the PCM "mi" as detailed in the Appendix; and Фmi is the complex phase function modeling MCF longitudinal random perturbations, temporal birefringence effects and optical attenuation: where α is the power attenuation coefficient of the MCF (assumed similar in each PCM), and ϕmi is the real phase function involving the longitudinal and temporal MCF perturbations: Here, βmi is the unperturbed phase constant of the polarized core mode "mi"; βmi (B) is the phase perturbation modeling macrobends; and βmi (S) is the phase perturbation including structural fluctuations induced by microbending, twisting, temporal fiber birefringence and additional temporal variations as the fluctuation of the state of polarization of the laser.It should be noted that the functions Ami, Fmi and Фmi also depend on the angular frequency ω0.However, considering monochromatic electric fields the frequency dependence has been omitted in the notation.Furthermore, as it was pointed out previously, the Fmi function is assumed longitudinal dependent.Considering that the phase function Фmi is modified along the MCF length, the transversal function Fmi should also be modified along the longitudinal direction of the fiber.Hence, assuming the function Fmi(x,y;z,t)•exp(−jФmi(z;t)) as local eigenmode in Δz ≥ λ, the following Helmholtz equation should be satisfied in each "mi" polarized core mode: Therefore, using Eq. ( 5) and assuming that ∂zFmi ≈ 0 and ∂z 2 Fmi ≈ 0 in Δz ≈ λ, we obtain the following relation: ; , ; , , where ΔT = ∂x 2 + ∂y 2 is the transverse Laplacian operator.A similar expression is found for the polarized core mode bi exchanging 'a' by 'b' in Eq. (10).Now, substituting the global electric field given by Eq. ( 7) into the nonlinear coupled wave equations Eqs.(4) using Eq. ( 10), considering the slowly varying envelope approximation with ∂z 2 Ami ≈ 0 in Δz ≈ λ, multiplying by the respective Fmi(x,y;z,t)•exp(+jϕmi(z;t)) and integrating in an infinite cross-section area of the MCF (as indicated in [22] for the classical derivation of the perturbation theory), we obtain the following coupled local-mode equation: where cax, max,ay, kax,bx, χax,bx, ηax,by, qax and gax are the mode-coupling coefficients (MCCs) defined in the next subsection; and the mismatching functions are defined as Δϕay,ax := ϕay−ϕax, Δϕbx,ax := ϕbx−ϕax and Δϕby,ax := ϕby−ϕax.It should be noted that the theoretical model is completed by three additional coupled local-mode equations for the ay, bx and by polarized core modes, which can be obtained just by exchanging the corresponding subindexes in Eq. (11).It is worth mentioning that additional nonlinear terms modeling cross-coupling effects should be included in Eq. ( 11) for coupled MCFs with a core pitch value (dab) lower than three times the core radius (R0).However, if we assume a MCF with dab >> 3R0, the self-coupling effect is the predominant nonlinear coupling effect and the additional nonlinear terms can be neglected [13].

Linear and nonlinear mode-coupling coefficients
The linear MCCs of the coupled local-mode equation Eq. ( 11) are defined as: ; , , ; with [29]: where fT is the twist rate along the MCF length; p11 and p12 are components of the photo-elastic tensor [30]; and na(b) is the average value of the material refractive index of each core excluding temporal birefringence fluctuation.It should be remarked that considering the temporal birefringence random fluctuation of na(b) of the order of ~10 −7 (as it is experimentally verified in Section 4), this fluctuation can be neglected when na(b) is raised to the fourth power in Eq. ( 13).In addition, the nonlinear MCCs are given by the expressions (in W −1 •m −1 ): , ; , ; , 2 ; ; where Nax(z;t):= ∫∫Fax 2 (x,y;z,t)dS∞; S∞ is the infinite cross-sectional area of the MCF; and Sa(b) is the cross-sectional area of each core.It should be noted that, in coherence with [18], the longitudinal random perturbations of the fiber are included in the MCCs.Nevertheless, in our method we have considered not only longitudinal perturbations, but also temporal ones.As a result, the derived MCCs are found to be time dependent.Moreover, note that our model inherently incorporates these stochastic perturbations, as they were directly included in the Maxwell equations.Specifically, they also appear in the exponential terms of Eq. (11), which account for possible slight changes in the longitudinal or temporal birefringent conditions.
The linear MCCs χax,bx, cax and kax,bx are similar to the linear MCCs investigated in [13], where it was observed that kax,bx is the predominant linear MCC and that χax,bx and cax can be neglected.However, if polarization effects are included in the Maxwell equations, two new linear MCCs appear, max,ay and ηax,by, which model the iC-XT and XIC-XT, respectively.In order to investigate whether these coefficients should be maintained in Eq. ( 11) when compared with kax,bx, the ratios max,ay/kax,bx and ηax,by/kax,bx are analyzed following a similar procedure to the one discussed in [13].
Assuming a straight MCF with constant twist rate fT and without temporal birefringence, the linear MCCs max,ay and ηax,by are found to be only longitudinal dependent and are maximized when considering homogeneous PCMs with nai = nbi (i = x,y) and Fax(bx) = Fay(by).As a result, the spatial average of the ratios <max,ay(z)/kax,bx(z)> and <ηax,by(z)/kax,bx(z)> can be calculated from Eqs. ( 12) with a similar derivation as in [13]: where R0 is the core radius, nc is the cladding refractive index and the modal parameters ua(b) and wa(b) are given by [23].As an example, the ratios given by Eqs. ( 16 Figure 2 shows the behavior of the calculated ratios for different constant twist rate values fT and different values of the ratio between the core pitch and the core radius dab/R0.From Fig. 2(a), it can be noted that max,ay should be maintained in Eq. (11), since this MCC is usually of the same order or higher than kax,bx.As expected, the higher the twist rate value fT, the higher the mode coupling between the x and y orthogonal polarizations in a given core.However, as depicted in Fig. 2(b), the linear MCC ηax,by can be neglected in Eq. ( 11) for any core pitch value.Hence, the XIC-XT will depend directly on the iC-XT of both cores, as verified in next sections.Note that the MCC analysis considering heterogeneous cores with na ≠ nb is not included in Fig. 2 because the conclusions in the predominant linear MCCs are found to be identical.
As a result, redefining the complex envelopes as and considering all adjacent cores to core a in a N-core MCF, the final expression of the coupled local-mode equation for the polarized core mode ax including linear and nonlinear intra-and inter-core crosstalk in MCF media with dab > 3R0 is given by the expression: Furthermore, considering non-monochromatic electric fields in Eqs. ( 1) and performing a similar mathematical discussion from the Maxwell equations, the CLMT can also be used to describe optical pulse propagation in SM-MCFs including the brackets of the left-hand side of Eq. ( 17) the dispersion operator , ˆt ax D defined as , where In addition, the inter-core skew and the additional dispersive effects inducing a crosstalk time dependence [6,27] can also be modeled via this dispersion operator.However, the fiber dispersion effects are omitted in this work in order to evaluate only the time dependence of the crosstalk induced by the temporal birefringence fluctuations of the MCF, as it was pointed out previously.Alternatively to the CLMT, the theoretical models reported in [14][15][16][17][18][19] can also be extended to describe the temporal perturbations of the optical media inducing a crosstalk time dependence in MMFs and MM-MCFs.

Equivalent refractive index model and numerical simulations
In this section, the MCF simulation model is detailed and extensive numerical simulations of the crosstalk mean between the PCMs are performed considering temporal and longitudinal birefringent effects of the optical media.We observe that the temporal birefringence fluctuation of each core modifies the average value of the iC-, DIC-and XIC-XT.In addition, fiber twisting is proposed as a strategy of birefringence management to balance the inter-core crosstalk between the PCMs of different cores.

Equivalent refractive-index model with longitudinal and temporal perturbations
In a given core m of a SM-MCF, the longitudinal and temporal perturbations of the MCF should be included in the equivalent refractive index nmi (eq) of each "mi" polarized core mode (LP01,mx and LP01,my).This index determines the value of the ϕmi and Fmi functions, as well as of the mode-coupling coefficients.In turn, these coefficients can be obtained via numerical integration of Eqs. ( 12) and (15) or through the closed-form expressions reported in [13] together with Eq. (16a) (substituting the material refractive index na by nax (eq) ).Furthermore, the ϕmi functions can be approximated in a short MCF segment, where the longitudinal MCF random perturbations are assumed to be approximately constant, as: eq eq eq 2 1 0 2 1 ; ; ; ; .
Hence, the ϕmi functions, MCCs and the coupled local-mode equations can be numerically solved by using the proper expression for nmi (eq) (z;t), which should include the longitudinal and temporal MCF perturbations.In [10], the equivalent refractive index model was proposed assuming a single polarization and without including temporal birefringence effects.Here, in order to simulate also a slowly varying longitudinal and temporal MCF birefringence fluctuation in a time period spanning several days, we extend the equivalent refractive index model of [10] as depicted in Fig. 3.
In particular, considering a MCF with lowly-birefringent (LB) or highly-birefringent (HB) cores, the Monte Carlo method with S random iterations can be performed in each PCM "mi", which is characterized in a given day j by the following parameters:  The Monte Carlo s-th iteration identified by the subindex s = 1,…,S. The number of segments Nm of the core m.Each core segment is identified by the subindex l = 1,…,Nm. The equivalent refractive index of the "mi" PCM (i = x,y) of the l-th segment associated with the s-th iteration for the j-th day: where nmi,j,s-l (t) describes the temporal birefringence fluctuation of the MCF given by a Gaussian random process as detailed later; θ0 is the offset of the twist angle of the MCF reference axis at z = 0; θm is the offset of the twist angle of core m measured from the MCF reference axis as shown in Fig. 3; dm is the Euclidean distance of core m to the MCF center; and RB,l and fT,l are the bending radius and twist rate in the l-th segment, respectively.It should be noted that the local orientation of the fiber principal axes (given by Eqs. ( 13) and ( 19)) is different in each segment as a consequence of the longitudinal changes in the twist rate between adjacent segments.On the other hand, although in the numerical simulations and the experimental work of the next sections we have not considered temporal changes in the bending radius and twist rate, in a real deployed MCF system the environmental factors could induce temporal fluctuations in the bending radius and twist rate of the optical media.In that case, RB,l and fT,l should be regarded as time-dependent variables.In addition, the temporal birefringence fluctuation |nmy,j,s-l(t)nmx,j,s-l(t)| of the MCF is modeled by a Gaussian random process N(μ(t),σ 2 ) with mean μ(t) = Δnm,j describing the value of the intrinsic linear birefringence in core m and day j, and variance σ 2 = δnm,l modeling the photo-elastic effect in the l-th segment induced by the macrobend RB,l as [31]: where nm is the average value of the material refractive index of core m given by Eq. ( 14) and dc is the MCF cladding diameter.Thus, the value nmi,j,s-l (t) can be calculated in the l-th segment and s-th iteration as (i = x,y):

Numerical simulations
The numerical analysis of the crosstalk between the PCMs was performed in line with the MCF simulation model detailed in previous subsection.Two different 150 m homogeneous two-core fibers (TCFs) comprising LB and HB cores were considered in the numerical simulations, as depicted in Fig. 4. The number of simulated segments and the average value of the linear birefringence in each core <Δnm,j> is indicated in Table 1.
We have assumed cores a and b with different birefringence average value and a different number of segments in the LB-TCF case, in order to compare the simulation results with the crosstalk behavior observed in the experimental measurements of the next section.As it is experimentally verified, the higher the birefringence average value <Δnm,j> is assumed in a given core m, the lower is the number of segments Nm that should be considered.Additional MCF parameters employed were: dab = 36 μm, dc = 125 μm, R0 = 4 μm, na = nb = 1.45, nc = 1.44, θ0 = 0, and λ =1550 nm.Although in a real LB-MCF the core radius R0 can also present longitudinal fluctuation, it is a less time-consuming approach to consider the longitudinal fluctuation of this fiber parameter as manufacturing imperfections encoded by <Δnm,j> in each core m = a,b.In HB-MCFs with elliptical cores, the intrinsic birefringence can also be included in <Δnm,j> assuming R0 as the average value of the minor and major axis dimensions.Figure 5 shows the simulation results of the iC-XT, DIC-XT and XIC-XT mean between the PCMs in the LB-TCF of Fig. 4(a) when varying the linear birefringence value Δna,j and Δnb,j throughout a 10-day period (j = 1,…,10).Following [12,13], the crosstalk mean from a PCM ξ to a PCM υ is defined as μυ,ξ := E[Pυ(z=L)/Pξ(z=L)], where E[•] is the expectation operator, L is the MCF length and (υ,ξ) ∈ {ax,ay,bx,by} 2 .Monte Carlo method was performed over 100 iterations (S = 100) considering a constant bending radius of RB = 100 cm along the MCF.In addition, two different twist rate average values of fT = 0.05 and 1 turns/m were assumed with Gaussian random fluctuations between adjacent segments around ~0.02 turns/m in both cases.The simulation results are shown in Figs.5(a1)−5(c1) and Figs.5(a2)−5(c2) for each twist rate average value, respectively.The temporal fluctuation of the linear birefringence Δnm,j (m=a,b) depicted in Figs.5(a1) and 5(a2) was calculated from a Gaussian distribution N(<Δnm,j>,10 −7 ) considering a similar evolution in both cores in coherence with the experimental results of the next section.
As we can notice from Figs. 5(a1) and 5(a2), the higher the linear birefringence Δnm,j, the lower the iC-XT mean observed in a given core m.The main reason for this comes from the higher phase mismatching between the intra-core PCMs LP01,mx and LP01,my.If the linear birefringence Δnm,j increases, the phase mismatching between both orthogonal polarizations increases reducing the iC-XT mean.Furthermore, increasing the fiber twisting from 0.05 to 1 turns/m, the iC-XT mean is also increased in both cores due to the increment of the circular birefringence, as it is observed when comparing the iC-XT levels in Figs.5(b1) and 5(b2).Note that, as the global birefringence (linear+circular) increases and prevails over the temporal random birefringence fluctuation (~10 −7 ), the temporal fluctuation of the iC-XT mean decreases.This is confirmed by the results for core b in Fig. 5(b2) (with a minimal excursion of 2 dB) and by the experimental measurements presented in the next section.
On the other hand, as depicted in Figs.5(c1) and 5(c2), the XIC-XT mean presents the same temporal evolution as the iC-XT mean indicating that XIC-XT is mainly generated by the intra-core mode coupling, as it was pointed out in Section 2. Thus, the XIC-XT mean is lower than the DIC-XT mean when the iC-XT mean of both cores is lower than 0 dB.However, increasing the fiber twisting from 0.05 to 1 turns/m, the iC-XT mean increases higher than 0 dB in core a the 1 st , 3 rd ,5 th , 6 th and 8 th days.As a result, the XIC-XT mean is higher than DIC-XT these days, as shown in Fig. 5(c2).1.The crosstalk behavior in the HB-TCF was also calculated over 100 iterations assuming the same bending and twisting conditions as in the LB-TCF.It should be noted from Figs. 6(a1) and 6(a2) that we cannot observe a significant temporal birefringence fluctuation, since the temporal random fluctuation of Δnm,j was assumed to be of the same order as in the LB cores (around 10 −7 ), much lower than the linear birefringence average value of <Δnm,j> = 4•10 −4 in the HB cores.Therefore, the temporal evolution of the iC-XT mean was found approximately similar in both cores with a maximal excursion of 1 dB, approximately.In addition, note that the iC-XT mean is much lower than in LB cores due to a higher mismatching between the refractive index of the intra-core PCMs.However, increasing the fiber twisting from 0.05 to 1 turns/m the iC-XT mean increases from −33 dB to −20 dB, as shown in Figs.6(b1) and 6(b2).Considering the low average value and temporal fluctuation of iC-XT in both cores a and b, the XIC-is lower than DIC-XT for both fT values, as shown in Figs.6(c1) and 6(c2).Nevertheless, when the average value of the fiber twist rate increases from 0.05 to 1 turns/m, the difference between DIC-and XIC-XT mean is reduced from +30 to +17 dB due to the increment of the iC-XT mean in both elliptical cores.
An additional comment on the results of Figs. 5 and 6 is in order.The mode coupling from the PCM ξ to PCM υ is the same as from υ to ξ, with (υ,ξ) ∈ {ax,ay,bx,by} 2  .This is to be expected when operating with a large bending radius RB, for which the condition Δϕξυ ≈ Δϕυξ can be inferred from Eqs. ( 18) and ( 19) for both LB-and HB-MCFs.
In order to further analyze the longitudinal MCF random perturbations induced by MCF bending and twisting, a multi-parameter simulation of the CLMT was performed over 80 Monte Carlo iterations considering a 2 m LB-TCF with cores a and b comprising a single segment with the same birefringence average value of <Δnaj> = <Δnbj> = 10 −7 . The temporal birefringence fluctuation of the TCF was omitted in this simulation.The numerical results are shown in Fig. 7. Figure 7 depicts the mean of: iC-XT ay-ax (similar to by-bx when assuming cores with similar birefringence), DIC-XT bx-ax and XIC-XT by-ax when changing the bending radius RB and fiber twisting conditions fT.As it can be noticed from Fig. 7(a), we cannot observe intra-core mode coupling between ax-ay with fT = 0 in 2 m of LB-TCF.Macrobending increases the phase mismatching between the PCMs ax and ay without inducing iC-XT due to the photo-elastic effect [31].As a result, significant XIC-XT cannot be observed for short MCF lengths when fT = 0, as in the case of Fig. 7(c).Nevertheless, an average level of DIC-XT between −100 and −50 dB can be noted from Fig. 7(b) in non-twisting conditions depending on the bending radius value.In addition, the higher the twist rate and the bending radius, the higher the iC-, DIC-and XIC-XT mean due to the reduction of the phase mismatching between the different PCMs of the TCF.Note that DIC-and XIC-XT means are balanced when the iC-XT mean achieves the value of 0 dB in Fig. 7(a).Therefore, MCF twisting can be proposed as a potential strategy for birefringence management to balance the inter-core crosstalk between the different PCMs for short MCF distances.For MCF distances of several kilometers, the iC-XT mean increases and the difference between the mean of the DIC-and XIC-XT are reduced.
Finally, we analyze the crosstalk behavior between the PCMs when operating in nonlinear regime.To this end, we simulated a LB-TCF with fiber parameters: L = 5 m, fT = 0.08, RB = 100 cm, Nm = 2 segments, S = 80 iterations, dab = 36 μm, dc = 125 μm, R0 = 4 μm, na = nb = 1.45, nc = 1.44, and λ =1550 nm, with a constant birefringence value in each core of Δna(b),j = 10 −7 (omitting temporal fluctuation).The power launch level injected into the PCM ax was increased from 0 to 40 dBm.The behavior of the mean of the iC-XT (ay-ax), DIC-XT (bx-ax) and XIC-XT (by-ax) is shown in Fig. 8.It should be noted from Fig. 8(a) that the iC-XT mean remains unchanged for a power launch level PL = Pax(z=0) lower than 40 dBm.Therefore, we conclude that the nonlinear birefringence induced by the nonlinear polarization P (3) in Eqs.(4) does not increase the iC-XT when considering realistic optical power levels.Moreover, as shown in Fig. 8(b), the mean of DIC-and XIC-XT is reduced as the power launch level increases.This is because the Kerr effect detunes the phase constants of PCMs between different cores, as reported in [12].Furthermore, the difference between the DIC-and XIC-XT mean is constant around 5 dB when the optical power launch increases, due to the constant behaviour of the iC-XT in the nonlinear regime.

Experimental results
In order to validate the theoretical analysis of the previous sections, experimental measurements are performed on a homogeneous four-core fiber (4CF) analyzing the temporal birefringence fluctuation of the optical media and its impact on the mean of the linear and nonlinear crosstalk between the PCMs.The employed experimental set-up is shown in Fig. 9.A tunable continuous-wave external cavity laser (CW-ECL) at 1550 nm with a linewidth of 50 kHz was used along with two polarization controllers (PC1 and PC2).The first polarization controller (PC1) was employed to align the polarization axes of the CW-ECL and the polarization beam splitter (PBS) in a back-to-back (B2B) connection, as detailed in Fig. 9.A polarization analyzer (Optellios PS2300) was used to verify that the output light of the PC1 was linearly polarized along the x axis by maximizing the upper branch of the PBS in the B2B connection.The second polarization controller (PC2) was inserted to modify the polarization launched into a given core of the 4CF.The additional erbium doped fiber amplifier (EDFA) followed by a variable optical attenuator (VOA) was employed only for the nonlinear crosstalk analysis between the PCMs.A 3D fan-in/fan-out device with 2.2 dB insertion losses was employed in order to inject and extract the optical power launched into a 150 m homogeneous single-mode 4CF Fibercore SM-4C1500(8.0/125)spooled on a reel with a constant bending radius of 100 cm and constant twist rate of 1 turns/m inside a methacrylate box to reduce the external perturbations.In addition, the laboratory room was isolated and the air cushions were inflated in our active optical table to minimize the external perturbations induced by human activity.

back-to-back connection
The crosstalk behavior between the PCMs of cores 1 and 3 was analyzed in the linear and nonlinear regime by injecting into a given PCM an optical power launch level of 0 dBm and 6 dBm, respectively.In the nonlinear regime, the EDFA gain was maximized in order to reduce the crosstalk averaging due to amplified spontaneous emission (ASE) noise, as recommended in [12], and the power launch was controlled by the VOA.The optical power of the two PCMs (LP01,mx and LP01,my) of a given core m (with m = 1,3) was separated with the PBS and measured with a power meter comprising two different channels (Thorlabs PM320E).
The mean of the linear and nonlinear iC-, DIC-and XIC-XT between the PCMs of cores 1 and 3 was estimated using the wavelength sweeping method [10,12] from 1540 to 1590 nm with 5 pm step.The CW-ECL was previously tested between 1540 and 1590 nm to confirm that the polarization remains unchanged when sweeping the laser wavelength.It should be noted that the wavelength sweeping method can also be used to estimate the value of the linear birefringence of each core in different days, as detailed in [32,33].Moreover, it should be remarked that the impact of the dispersive effects and the inter-core skew effect on the time dependence of the crosstalk [6,27] was minimized using a CW-ECL with a reduced linewidth of 50 kHz.As it was experimentally verified, considering that the linear birefringence of each core remains unchanged during more than 10 hours, we analyzed the temporal birefringence fluctuation by performing measurements at different days and months.Figure 10 shows the temporal fluctuation of the linear birefringence and the crosstalk mean behavior between the PCMs of cores 1 and 3 measured in six consecutive days in October 2015, four consecutive days in January 2016, and five consecutive days in March 2016, with similar temperature conditions in the laboratory set-up.As shown in Fig. 10(a), cores 1 and 3 present a different average value of the linear birefringence estimated to be <Δn1,j> = 4.9•10 −7 and <Δn3,j> = 1.2•10 −6 , respectively.It can be noted that the average value of the linear birefringence is constant in each core the three different measured months.Moreover, although the average value of the linear birefringence is different in each core, the temporal evolution of the linear birefringence presents a similar shape in both cores, in line with our assumptions in Section 3.
As depicted in Figs.10(a) and (b), the higher the linear birefringence in a given core, the lower the mean of the iC-XT.Furthermore, it should be noted that iC-XT in core 3 is lower than in core 1 due to a higher mismatching between the orthogonal polarizations.In line with the simulations of Section 3, the iC-XT mean presents a lower temporal fluctuation in the more birefringent core (core 3), which occurs when the average value of the birefringence is higher than the temporal random birefringence fluctuation (~10 −7 ).In addition, we can observe from Fig. 10(c) that the temporal evolution of the XIC-XT mean presents the same behavior as the iC-XT mean, indicating that XIC-XT depends directly on the iC-XT of both cores.As a result, DIC-XT is higher than XIC-XT when iC-XT is lower than 0 dB in both cores, as it was observed the 3 rd and 4 th days (October 2015), the 7 th day (January 2016), and the 13 th and 14 th days (March 2016).Furthermore, the experimental measurements of Fig. 10 fit correctly with the CLMT simulation when using Eqs.( 17)-( 21) with the parameters indicated in Table 1 for the LB-TCF with N1 = Na = 8, N3 = Nb = 6 and assuming a temporal birefringence Δn1,j = Δna,j and Δn3,j = Δnb,j as depicted in Fig. 10(a).Therefore, we conclude that the more birefringent a given core is, the lower number of segments should be used to simulate the crosstalk between the PCMs, as was pointed out in Section 3.
On the other hand, the nonlinear crosstalk between de PCMs 1y-1x, 3y-3x, 3x-1x and 3y-1x was also measured the 5 th , 6 th and 15 th days with a power level of 6 dBm launched into the corresponding PCM, taking into account the insertion losses of 2.2 dB of the 3D fan-in device.As it was previously observed in Section 3, the iC-XT mean presents the same value in nonlinear regime considering that the nonlinear birefringence can be neglected for realistic optical power launch levels lower than 40 dBm.In addition, the DIC-and XIC-XT mean is reduced around 1 dB keeping constant the difference between both inter-core crosstalk types as a direct consequence of the constant behavior of the iC-XT mean in nonlinear regime.
Moreover, although the crosstalk analysis between the PCMs is restricted only to cores 1 and 3, the linear birefringence Δnm,j was also measured in cores 2 and 4. The average value of the linear birefringence of cores 2 and 4 was found to be <Δn2,j> ≈ <Δn4,j> ≈ 7•10 −7 .In addition, the average value of the iC-XT mean of each core was analyzed as a function of the average value of the linear birefringence <Δnm,j> including a numerical simulation of the CLMT. Figure 11 shows the experimental and simulated results of the iC-XT mean evolution when increasing the average value of the linear birefringence <Δnm,j> in a given core using Eqs.( 17)-( 21) considering 80 Monte Carlo iterations.It was assumed a core m with the fiber parameters of the 4CF Fibercore SM-4C1500(8.0/125):nm = 1.452, nc = 1.444, λ = 1550 nm, dab = 36 μm, dc = 125 μm.The distance to the reference center of the 4CF is fixed at dm = dab/√2 = 25.46 μm.The bending radius and the twist rate was assumed with a similar value as in the experimental set-up with RB = 100 cm and fT = 1 turns/m, respectively.It can be noted that the iC-XT mean is reduced when the average value of the linear birefringence <Δnm,j> increases inducing a higher phase mismatching between orthogonal polarizations.Furthermore, it was confirmed that the simulated results fit correctly with the experimental measurements of each core, validating the theoretical analysis reported in previous sections.
The experimental analysis reported in this section addresses a non-real deployed SM-MCF system with constant bending and twisting conditions.For real deployed MCF systems, the results herein presented should be refined by considering the multimode regime, higher fiber distances and random bending radius and twist rate along the fiber length.Furthermore, the polarization-mode dispersion (PMD) and the inter-core skew should also be investigated in real deployed MCF systems.To this end, distributed and lumped analytical models can be proposed in MCF media as in standard single-mode systems [34,35].Alternatively, the CLMT can also be employed to evaluate the impact of the longitudinal and temporal fiber perturbations on the statistics of the PMD.

Conclusions
In this paper, we have reported the theoretical and experimental analysis of the intra-and inter-core crosstalk behavior in SM-MCF including the temporal and longitudinal birefringence perturbations of the optical media.In order to propose an accurate analytical model, the coupled local-mode theory was derived from the Maxwell equations including both temporal and longitudinal MCF birefringence perturbations in the phase functions and MCCs of the polarized core modes.Two new MCCs max,ay (z;t) and ηax,by (z;t) were found modeling iC-XT and XIC-XT, respectively.The theoretical analysis shows that ηax,by coefficient can be neglected indicating that the XIC-XT mean depends directly on the iC-XT mean observed in both cores a and b, as it was verified numerically and experimentally.
In addition, a MCF simulation model of the CLMT was proposed and validated experimentally including both longitudinal and temporal birefringent effects in the equivalent refractive indexes nmi (eq) .The numerical simulations comprising LB and HB cores indicate that the higher is the linear birefringence in a given core, the lower is the value and the temporal fluctuation of the iC-XT mean.Furthermore, the XIC-XT mean presents the same temporal evolution as the iC-XT mean indicating that XIC-XT is mainly generated due to intra-core mode coupling.As a result, the XIC-XT mean is lower than the DIC-XT mean when the iC-XT mean of both cores is lower than 0 dB.Both DIC-and XIC-XT are balanced when iC-XT mean achieves the value of 0 dB in each core.Hence, MCF twisting was proposed as a birefringence management strategy in short MCF distances to balance the DICand XIC-XT.However, when higher MCF distances of several kilometers are considered, the iC-XT mean increases reducing the difference between the mean of the DIC-and XIC-XT.In nonlinear regime it was observed that the nonlinear iC-XT mean remains unchanged when the power level launched into a given polarized core mode increases.In contrast, the DIC-and XIC-XT mean is reduced in nonlinear regime showing a similar evolution.Finally, extensive experimental measurements performed in three different months using a homogeneous 4CF confirm the crosstalk behavior observed in the numerical simulations between the PMCs.The experimental validation also pointed out that the temporal linear birefringence evolution is similar in different cores of the optical media.

Appendix: calculation of the transversal local eigenfunctions Fmi(x,y;z,t)
The transversal local eigenfunction Fmi for the PCM "mi" can be calculated using the closed-form expressions detailed in [23] for the LP01 mode in the cores and cladding regions.However, considering that the local eigenfunction depends on the longitudinal and temporal MCF random perturbations, it should be calculated as a function of the equivalent refractive index nmi (eq) detailed in Section 3. Therefore, the eigenfunction Fmi(r;z,t) can be expressed in a local polar coordinate system (r,φ,z) of a given core m as: where r = (x 2 +y 2 ) 1/2 ; R0 is the core radius (assuming all cores of the MCF with identical radius); umi and wmi are the modal parameters of the PCM "mi", which can be calculated as: ; ; , with nmi (eq) given by Eq. ( 19), CF an auxiliary parameter referred as the correction factor and nc the material refractive index of the cladding.The correction factor CF is included in Eq. ( 24) to calculate the correct values of the normalized frequency when using the nominal value of the core radius.It is worth mentioning that in real SMFs and MCFs, the core radius presents longitudinal variations along the fiber length, as was pointed out previously in Section 3. In order to consider these core radius fluctuations, we have included in Eq. ( 24) the correction factor CF, which should be calculated using the cut-off wavelength (λC) of the fiber as a reference.Specifically, the correction factor is found as CF = 0.7 in Section 3.2 and CF = 0.9 in Section 4 (λC = 1410 nm).Moreover, note that in Eq. ( 24) it was assumed the same material refractive index of the cladding for the x and y polarization axes.

Fig. 1 :
Fig. 1: Multi-core fiber comprising different birefringent segments in cores a and b with longitudinal and temporal varying fluctuations in the refractive index tensor.

Fig. 5 .Fig. 6 .
Fig. 5. Simulation results of the crosstalk behavior between the polarized core modes in a 150 m two-core fiber with lowly-birefringent cores, considering the temporal fluctuation of the linear birefringence of each core with a twist rate value of fT = 0.05 and 1 turns/m: (a) Simulated linear birefringence evolution in cores a and b throughout a 10-day period; (b) intra-core crosstalk evolution; and (c) direct and cross inter-core crosstalk evolution.(iC-XT: intra-core crosstalk, IC-XT: inter-core crosstalk).

Figure 6
Figure 6 shows the simulated results considering the 150 m HB-TCF depicted in Fig. 4(b) with an average value of the linear birefringence of <Δnm,j> = 4•10 −4in both cores (m = a,b), as indicated in Table1.The crosstalk behavior in the HB-TCF was also calculated over 100 iterations assuming the same bending and twisting conditions as in the LB-TCF.Figures 6(a1)−6(c1) and Figs.6(a2)−6(c2) show the simulated results for twist rate average values of fT = 0.05 and 1 turns/m, respectively.It should be noted from Figs. 6(a1) and 6(a2) that we cannot observe a significant temporal birefringence fluctuation, since the temporal random fluctuation of Δnm,j was

Figures 6 (
Figure 6 shows the simulated results considering the 150 m HB-TCF depicted in Fig. 4(b) with an average value of the linear birefringence of <Δnm,j> = 4•10 −4in both cores (m = a,b), as indicated in Table1.The crosstalk behavior in the HB-TCF was also calculated over 100 iterations assuming the same bending and twisting conditions as in the LB-TCF.Figures 6(a1)−6(c1) and Figs.6(a2)−6(c2) show the simulated results for twist rate average values of fT = 0.05 and 1 turns/m, respectively.It should be noted from Figs. 6(a1) and 6(a2) that we cannot observe a significant temporal birefringence fluctuation, since the temporal random fluctuation of Δnm,j was

Fig. 7 .
Fig. 7. Multi-parameter simulation of the crosstalk between polarized core modes varying the bending radius (1/RB) and the twist rate (fT) in a 2 m homogenous two-core fiber with lowly-birefringent cores.(a) intra-core crosstalk mean ay-ax, (b) direct inter-core crosstalk mean bx-ax, and (c) cross inter-core crosstalk mean by-ax.

Fig. 8 .
Fig.8.Numerical results of the nonlinear crosstalk evolution considering a theoretical range of the power launch level in the ax polarized core mode in a homogeneous 5 m lowly-birefringent two-core fiber: (a) nonlinear intra-core crosstalk mean between ay-ax, and (b) nonlinear direct and cross inter-core crosstalk mean between bx-ax and by-ax.(iC-XT: intra-core crosstalk, DIC-XT: direct inter-core crosstalk, XIC-XT: cross inter-core crosstalk).

1 Fig. 9 .
Fig.9.Experimental set-up for intra-and inter-core crosstalk evaluation between the polarized core modes of cores 1 and 3 of a four-core fiber (4CF), considering multi-core fiber temporal birefringence fluctuation and both linear and nonlinear power regimes.

Fig. 10 :
Fig.10: Experimental results of the temporal linear birefringence fluctuation over different days and months of a 150 m 4CF, and corresponding intra-and inter-core crosstalk mean between cores 1 and 3. A power launch level of PL = 0 dBm was used in linear regime (NL: nonlinear regime with PL = 6 dBm).(a) Linear birefringence evolution of cores 1 and 3, (b) intra-core crosstalk 1x-1y and 3x-3y, and (c) direct-and cross-inter-core crosstalk 1i-3j with i,j = x,y

4 Fig. 11 :
Fig.11: Experimental measurement and simulated results of the intra-core crosstalk (iC-XT) mean evolution with the average value of the linear birefringence.Solid points: measured data of cores 1, 2, 3, 4 in a 150 m homogeneous four-core fiber.Dashed line: simulation results using the coupled-local mode equation, Eq. (17).