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

Andrés Macho; Carlos García-Meca; F. Javier Fraile-Peláez; Maria Morant; Roberto Llorente

doi:10.1364/OE.24.021415

1. 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 [1–3]. 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 [8–13] 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 [14–17] 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 [8–19], 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 [5–7] 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 [7–13] 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 highly-birefringent (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.

2. 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 χ_ij⁽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 LP₀₁_x and LP₀₁_y 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.

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.

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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 local modes where the longitudinal and temporal birefringence conditions are approximately constant.

In this way, in contrast with [4–13] 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.

2.1 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 electric field (Ɛ), the linear polarization (Ƥ⁽¹⁾) and the nonlinear polarization (Ƥ⁽³⁾) associated with the m-th core of a weakly guiding MCF can be written as [24–26]:

\vec{ε} (\vec{r}, t) = \sum_{i = x, y} \frac{1}{2} [E_{i, ω_{0}} (\vec{r}; t) \exp (j ω_{0} t) + c . c .] {\hat{u}}_{i},

{\vec{P}}^{(1)} (\vec{r}, t) = \sum_{i = x, y} \frac{1}{2} [P_{i, ω_{0}}^{(1)} (\vec{r}; t) \exp (j ω_{0} t) + c . c .] {\hat{u}}_{i},

{\vec{P}}^{(3)} (\vec{r}, t) \approx \sum_{i = x, y} \frac{1}{2} [P_{i, ω_{0}}^{(3)} (\vec{r}; t) \exp (j ω_{0} t) + c . c .] {\hat{u}}_{i},

where E_i,ω₀, P_i,ω₀⁽¹⁾ and P_i,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 P_i,₃_ω₀⁽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]:

P_{i, ω_{0}}^{(1)} (\vec{r}; t) = ε_{0} χ_{i j}^{(1)} (z; t) E_{j, ω_{0}} (\vec{r}; t),

P_{i, ω_{0}}^{(3)} (\vec{r}; t) = \frac{3}{4} ε_{0} χ_{i j k l}^{(3)} (- ω_{0}; ω_{0}, ω_{0}, - ω_{0}) E_{j, ω_{0}} (\vec{r}; t) E_{k, ω_{0}} (\vec{r}; t) E_{l, - ω_{0}} (\vec{r}; t),

where the first-order electrical susceptibility tensor χ_ij⁽¹⁾ is assumed longitudinal and temporal dependent due to the MCF perturbations. Moreover, the third-order susceptibility χ_ijkl ⁽³⁾ can be approximated in silica media to [26]:

χ_{i j k l}^{(3)} (- ω_{0}; ω_{0}, ω_{0}, - ω_{0}) = \frac{1}{3} χ_{x x x x}^{(3)} (- ω_{0}; ω_{0}, ω_{0}, - ω_{0}) \cdot (δ_{i j} δ_{k l} + δ_{i k} δ_{j l} + δ_{i l} δ_{j k}),

where χ_xxxx ⁽³⁾ = χ_ijkl ⁽³⁾ with i = j = k = l = x and δ_ij is the Kronecker delta function defined as δ_ij = 1 when i = j and δ_ij = 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:

\begin{array}{l} Δ E_{x, ω_{0}} (\vec{r}; t) + k_{0}^{2} [ε_{r, x} (\vec{r}; t) E_{x, ω_{0}} (\vec{r}; t) + σ (\vec{r}; t) E_{y, ω_{0}} (\vec{r}; t)] \\ + k_{0}^{2} γ [({| E_{x, ω_{0}} (\vec{r}; t) |}^{2} + \frac{2}{3} {| E_{y, ω_{0}} (\vec{r}; t) |}^{2}) E_{x, ω_{0}} (\vec{r}; t) + \frac{1}{3} E_{y, ω_{0}}^{2} (\vec{r}; t) E_{x, - ω_{0}} (\vec{r}; t)] = 0, \end{array}

\begin{array}{l} Δ E_{y, ω_{0}} (\vec{r}; t) + k_{0}^{2} [ε_{r, y} (\vec{r}; t) E_{y, ω_{0}} + σ (\vec{r}; t) E_{x, ω_{0}}] \\ + k_{0}^{2} γ [({| E_{y, ω_{0}} (\vec{r}; t) |}^{2} + \frac{2}{3} {| E_{x, ω_{0}} (\vec{r}; t) |}^{2}) E_{y, ω_{0}} (\vec{r}; t) + \frac{1}{3} E_{x, ω_{0}}^{2} (\vec{r}; t) E_{y, - ω_{0}} (\vec{r}; t)] = 0, \end{array}

where Δ is the Laplacian operator; ε_r,i = 1 + χ_ii⁽¹⁾; σ = χ_xy⁽¹⁾; γ = 0.75·χ_xxxx⁽³⁾; k₀ = ω₀/c₀ is the wavenumber at the vacuum and E_i,−ω₀ = (E_i,ω₀)*, where * denotes complex conjugation. Note that we have assumed in Eq. (4b) that χ_xy⁽¹⁾ = χ_yx⁽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 ε_r,i is defined in each polarization axis i = x,y as (we use the symbol: = to indicate definition):

\begin{array}{l} ε_{r, i} (\vec{r}; t) : = ε_{r, c i} (\vec{r}; t) + Δ ε_{r, a i} (\vec{r}; t) + Δ ε_{r, b i} (\vec{r}; t) = \\ = {\begin{matrix} \vec{r} \equiv core a : ε_{r, a i} (z; t) = ε_{r, c i} (z; t) + Δ ε_{r, a i} (z; t) = n_{a i}^{2} (z; t) \\ \vec{r} \equiv cladding: ε_{r, c i} (z; t) = n_{c i}^{2} (z; t) \\ \vec{r} \equiv core b : ε_{r, b i} (z; t) = ε_{r, c i} (z; t) + Δ ε_{r, b i} (z; t) = n_{b i}^{2} (z; t) \end{matrix} \end{array}

where ε_r,ai, ε_r,bi and ε_r,ci are the relative electric permittivity in the cores a, b and in the cladding, respectively; Δε_r,ai and Δε_r,bi are the difference between the relative electric permittivity of the cladding and those of cores a and b, respectively; and n_ai, n_bi and n_ci 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 ε_r,x ≠ ε_r,y and the circular birefringence is given by:

σ (\vec{r}; t) : = {\begin{matrix} \vec{r} \equiv core a : σ_{a} (z; t) = χ_{a, x y}^{(1)} (z; t) \\ \vec{r} \equiv cladding: σ_{c} (z; t) = χ_{c, x y}^{(1)} (z; t) \\ \vec{r} \equiv core b : σ_{b} (z; t) = χ_{b, x y}^{(1)} (z; t) \end{matrix}

Furthermore, considering the low nonlinear nature of silica media, the nonlinear parameter γ is assumed constant in each dielectric region of the MCF.

2.2 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):

E_{i, ω_{0}} (\vec{r}; t) \approx \sum_{m = a, b} E_{m i, ω_{0}} (\vec{r}; t) = \sum_{m = a, b} A_{m i} (z; t) F_{m i} (x, y; z, t) \exp [- j Φ_{m i} (z; t)],

where E_mi is the complex amplitude of the electric field of the PCM “mi” considering isolated cores; A_mi is the complex envelope; F_mi is the transversal local eigenfunction which can be calculated using the equivalent refractive index n_mi^(eq) of the PCM “mi” as detailed in the Appendix; and Ф_mi is the complex phase function modeling MCF longitudinal random perturbations, temporal birefringence effects and optical attenuation:

Φ_{m i} (z; t) : = ϕ_{m i} (z; t) - j \frac{1}{2} α z,

where α is the power attenuation coefficient of the MCF (assumed similar in each PCM), and ϕ_mi is the real phase function involving the longitudinal and temporal MCF perturbations:

ϕ_{m i} (z; t) : = \int_{0}^{z} β_{m i}^{(eq)} (δ; t) d δ = β_{m i} z + \int_{0}^{z} β_{m i}^{(B)} (δ; t) + β_{m i}^{(S)} (δ; t) d δ .

Here, β_mi is the unperturbed phase constant of the polarized core mode “mi”; β_mi^(B) is the phase perturbation modeling macrobends; and β_mi^(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 A_mi, F_mi and Ф_mi 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 F_mi function is assumed longitudinal dependent. Considering that the phase function Ф_mi is modified along the MCF length, the transversal function F_mi should also be modified along the longitudinal direction of the fiber. Hence, assuming the function F_mi(x,y;z,t)·exp(−jФ_mi(z;t)) as local eigenmode in Δz ≥ λ, the following Helmholtz equation should be satisfied in each “mi” polarized core mode:

Δ [F_{m i} (x, y; z, t) \exp (- j Φ_{m i} (z; t))] + k_{0}^{2} ε_{r, m i} (\vec{r}; t) [F_{m i} (x, y; z, t) \exp (- j Φ_{m i} (z; t))] = 0;

Therefore, using Eq. (5) and assuming that ∂_zF_mi ≈0 and ∂_z²F_mi ≈0 in Δz ≈λ, we obtain the following relation:

\begin{array}{l} Δ_{T} F_{a i} (x, y; z, t) + k_{0}^{2} ε_{r, i} (\vec{r}; t) F_{a i} (x, y; z, t) = \\ = [k_{0}^{2} Δ ε_{r, b i} (\vec{r}; t) + j \partial_{z} β_{a i}^{(eq)} (z; t) + {(β_{a i}^{(eq)} (z; t))}^{2} - j α β_{a i}^{(eq)} (z; t)] F_{a i} (x, y; z, t), \end{array}

where Δ_T = ∂_x² + ∂_y² 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 ∂_z²A_mi ≈0 in Δz ≈λ, multiplying by the respective F_mi(x,y;z,t)·exp( + jϕ_mi(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:

\begin{array}{l} j \partial_{z} A_{a x} (z; t) = c_{a x} (z; t) A_{a x} (z; t) + m_{a x, a y} (z; t) \exp (- j Δ ϕ_{a y, a x} (z; t)) A_{a y} (z; t) \\ + \exp (- j Δ ϕ_{b x, a x} (z; t)) [k_{a x, b x} (z; t) - j χ_{a x, b x} (z; t) \partial_{z}] A_{b x} (z; t) \\ + η_{a x, b y} (z; t) \exp (- j Δ ϕ_{b y, a x} (z; t)) A_{b y} (z; t) \\ + \exp (- α z) [q_{a x} (z; t) {| A_{a x} (z; t) |}^{2} + g_{a x} (z; t) {| A_{a y} (z; t) |}^{2}] A_{a x} (z; t) \\ + \frac{1}{2} \exp (- α z) g_{a x} (z; t) \exp (- j 2 Δ ϕ_{a y, a x} (z; t)) A_{a x}^{*} (z; t) A_{a y}^{2} (z; t), \end{array}

where c_ax, m_ax,ay, k_ax,bx, χ_ax,bx, η_ax,by, q_ax and g_ax are the mode-coupling coefficients (MCCs) defined in the next subsection; and the mismatching functions are defined as Δϕ_ay,ax : = ϕ_ay−ϕ_ax, Δϕ_bx,ax : = ϕ_bx−ϕ_ax and Δϕ_by,ax : = ϕ_by−ϕ_ax. 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 (d_ab) lower than three times the core radius (R₀). However, if we assume a MCF with d_ab >> 3R₀, the self-coupling effect is the predominant nonlinear coupling effect and the additional nonlinear terms can be neglected [13].

2.3 Linear and nonlinear mode-coupling coefficients

The linear MCCs of the coupled local-mode equation Eq. (11) are defined as:

χ_{a x, b x} (z; t) : = \frac{β_{b x}^{(eq)} (z; t)}{β_{a x}^{(eq)} (z; t) N_{a x} (z; t)} \iint F_{a x} (x, y; z, t) F_{b x} (x, y; z, t) d S_{\infty},

c_{a x} (z; t) : = \frac{k_{0}^{2}}{2 β_{a x}^{(eq)} (z; t) N_{a x} (z; t)} \iint Δ ε_{r, b x} (z; t) F_{a x}^{2} (x, y; z, t) d S_{\infty},

m_{a x, a y} (z; t) : = \frac{k_{0}^{2}}{2 β_{a x}^{(eq)} (z; t) N_{a x} (z; t)} \iint σ (\vec{r}; t) F_{a x} (x, y; z, t) F_{a y} (x, y; z, t) d S_{\infty},

k_{a x, b x} (z; t) : = \frac{k_{0}^{2}}{2 β_{a x}^{(eq)} (z; t) N_{a x} (z; t)} \iint Δ ε_{r, a x} (z; t) F_{a x} (x, y; z, t) F_{b x} (x, y; z, t) d S_{\infty},

\begin{array}{l} η_{a x, b y} (z; t) : = \frac{k_{0}^{2}}{2 β_{a x}^{(eq)} (z; t) N_{a x} (z; t)} \iint σ (\vec{r}; t) F_{a x} (x, y; z, t) F_{b y} (x, y; z, t) d S_{\infty} \\ \approx \frac{k_{0}^{2}}{2 β_{a x}^{(eq)} (z; t) N_{a x} (z; t)} [\begin{array}{l} σ_{b} (z; t) \iint F_{a x} (x, y; z, t) F_{b y} (x, y; z, t) d S_{b} \\ + σ_{a} (z; t) \iint F_{a x} (x, y; z, t) F_{b y} (x, y; z, t) d S_{a} \end{array}], \end{array}

with [29]:

σ_{a (b)} (z) \approx π R_{0} f_{T} (z) n_{a (b)}^{4} | p_{11} - p_{12} |,

n_{a (b)} = (n_{a x (b x)} + n_{a y (b y)}) / 2,

where f_T is the twist rate along the MCF length; p₁₁ and p₁₂ are components of the photo-elastic tensor [30]; and n_a₍_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 n_a₍_b₎ of the order of ~10⁻⁷ (as it is experimentally verified in Section 4), this fluctuation can be neglected when n_a₍_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):

q_{a x} (z; t) : = \frac{k_{0}^{2}}{2 β_{a x}^{(eq)} (z; t) N_{a x} (z; t)} \iint γ F_{a x}^{4} (x, y; z, t) d S_{\infty},

g_{a x} (z; t) : = \frac{k_{0}^{2}}{3 β_{a x}^{(eq)} (z; t) N_{a x} (z; t)} \iint γ F_{a x}^{2} (x, y; z, t) F_{a y}^{2} (x, y; z, t) d S_{\infty},

where N_ax(z;t): = ∫∫F_ax²(x,y;z,t)dS_∞; S_∞ is the infinite cross-sectional area of the MCF; and S_a₍_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 χ_ax,bx, c_ax and k_ax,bx are similar to the linear MCCs investigated in [13], where it was observed that k_ax,bx is the predominant linear MCC and that χ_ax,bx and c_ax can be neglected. However, if polarization effects are included in the Maxwell equations, two new linear MCCs appear, m_ax,ay and η_ax,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 k_ax,bx, the ratios m_ax,ay/k_ax,bx and η_ax,by/k_ax,bx are analyzed following a similar procedure to the one discussed in [13].

Assuming a straight MCF with constant twist rate f_T and without temporal birefringence, the linear MCCs m_ax,ay and η_ax,by are found to be only longitudinal dependent and are maximized when considering homogeneous PCMs with n_ai = n_bi (i = x,y) and F_ax₍_bx₎ = F_ay₍_by₎. As a result, the spatial average of the ratios <m_ax,ay(z)/k_ax,bx(z)> and <η_ax,by(z)/k_ax,bx(z)> can be calculated from Eqs. (12) with a similar derivation as in [13]:

〈 \frac{m_{a x, a y} (z)}{k_{a x, b x} (z)} 〉 \approx \frac{σ_{a}}{2 (n_{a}^{2} - n_{c}^{2})} \frac{u_{a} [J_{0}^{2} (u_{a}) + J_{1}^{2} (u_{a})]}{J_{0} (u_{a}) J_{1} (u_{a})} \frac{K_{0} (w_{b})}{K_{0} (w_{b} d_{a b} / R_{0})},

〈 \frac{η_{a x, b y} (z)}{k_{a x, b x} (z)} 〉 \approx \frac{1}{(n_{a}^{2} - n_{c}^{2})} [σ_{b} \frac{u_{a}}{u_{b}} \frac{J_{0} (u_{a})}{J_{0} (u_{b})} \frac{J_{1} (u_{b})}{J_{1} (u_{a})} \frac{K_{0} (w_{b})}{K_{0} (w_{a})} \frac{K_{0} (w_{a} d_{a b} / R_{0})}{K_{0} (w_{b} d_{a b} / R_{0})} + σ_{a}] \approx \frac{σ_{b} + σ_{a}}{n_{a}^{2} - n_{c}^{2}},

where R₀ is the core radius, n_c is the cladding refractive index and the modal parameters u_a₍_b₎ and w_a₍_b₎ are given by [23]. As an example, the ratios given by Eqs. (16) are calculated assuming an ideal straight homogeneous two-core fiber with parameters: n_a₍_b₎ = 1.45, n = 1.44, λ = 1550 nm, R₀ = 4 μm.

Figure 2 shows the behavior of the calculated ratios for different constant twist rate values f_T and different values of the ratio between the core pitch and the core radius d_ab/R₀. From Fig. 2(a), it can be noted that m_ax,ay should be maintained in Eq. (11), since this MCC is usually of the same order or higher than k_ax,bx. As expected, the higher the twist rate value f_T, 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 η_ax,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 n_a ≠ n_b is not included in Fig. 2 because the conclusions in the predominant linear MCCs are found to be identical.

Fig. 2 Comparison of the linear mode-coupling coefficients (MCCs) m_ax,ay and η_ax,by with the linear MCC k_ax,bx using the spatial average ratios given by Eqs. (16) for different constant twist rate values f_T and core pitch ratio d_ab/R₀: (a) <m_ax,ay/k_ax,bx> and (b) <η_ax,by/k_ax,bx>. (Legend shown on Fig. 2(b) also applies to Fig. 2(a)).

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As a result, redefining the complex envelopes as $A (z; t) : = A (z; t) \exp (α z / 2)$ 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 d_ab > 3R₀ is given by the expression:

\begin{array}{l} j (\partial_{z} + \frac{α}{2}) A_{a x} (z; t) = m_{a x, a y} (z; t) \exp (- j Δ ϕ_{a y, a x} (z; t)) A_{a y} (z; t) \\ + \sum_{δ = b}^{N} k_{a x, δ x} (z; t) \exp (- j Δ ϕ_{δ x, a x} (z; t)) A_{δ x} (z; t) \\ + (q_{a x} (z; t) {| A_{a x} (z; t) |}^{2} + g_{a x} (z; t) {| A_{a y} (z; t) |}^{2}) A_{a x} (z; t) \\ + \frac{1}{2} g_{a x} (z; t) \exp (- j 2 Δ ϕ_{a y, a x} (z; t)) A_{a x}^{*} (z; t) A_{a y}^{2} (z; t) . \end{array}

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 inside the brackets of the left-hand side of Eq. (17) the dispersion operator ${\hat{D}}_{t, a x}$ defined as ${\hat{D}}_{t, a x} : = \sum_{k = 1}^{\infty} {(- j)}^{k - 1} β_{a x}^{(k)} \partial_{t}^{k} / k!$ , where $β_{a x}^{(k)} : = \partial_{ω}^{k} β_{a x} (ω_{0})$ . 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 [14–19] can also be extended to describe the temporal perturbations of the optical media inducing a crosstalk time dependence in MMFs and MM-MCFs.

3. 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.

3.1 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 n_mi^(eq) of each “mi” polarized core mode (LP_01,_mx and LP_01,_my). This index determines the value of the ϕ_mi and F_mi 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 n_a by n_ax^(eq)). Furthermore, the ϕ_mi functions can be approximated in a short MCF segment, where the longitudinal MCF random perturbations are assumed to be approximately constant, as:

ϕ_{m i} (z; t) = \int_{z_{1}}^{z_{2}} β_{m i}^{(eq)} (ξ; t) d ξ \approx β_{m i}^{(eq)} (z; t) \cdot (z_{2} - z_{1}) = k_{0} n_{m i}^{(eq)} (z; t) \cdot (z_{2} - z_{1}) .

Hence, the ϕ_mi functions, MCCs and the coupled local-mode equations can be numerically solved by using the proper expression for n_mi^(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.

Fig. 3 Multi-core fiber simulation model including longitudinal perturbations and temporal birefringence effects.

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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 N_m of the core m. Each core segment is identified by the subindex l = 1,…,N_m.
• 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: $n_{m i, j, s - l}^{(eq)} (z; t) = n_{m i, j, s - l} (t) [1 + \frac{d_{m}}{R_{B, l}} \cos (2 π f_{T, l} z + θ_{0} + θ_{m})],$
where n_mi,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; θ_m is the offset of the twist angle of core m measured from the MCF reference axis as shown in Fig. 3; d_m is the Euclidean distance of core m to the MCF center; and R_B_,l and f_T_,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, R_B_,l and f_T_,l should be regarded as time-dependent variables. In addition, the temporal birefringence fluctuation |n_my,j,s-l(t) – n_mx,j,s-l(t)| of the MCF is modeled by a Gaussian random process N(μ(t),σ²) with mean μ(t) = Δn_m,j describing the value of the intrinsic linear birefringence in core m and day j, and variance σ² = δn_m,l modeling the photo-elastic effect in the l-th segment induced by the macrobend R_B_,l as [31]:
$δ n_{m, l} \approx 0.011 n_{m}^{3} d_{c}^{2} / R_{B, l}^{2},$
where n_m is the average value of the material refractive index of core m given by Eq. (14) and d_c is the MCF cladding diameter. Thus, the value n_mi,j,s-l (t) can be calculated in the l-th segment and s-th iteration as (i = x,y):
$n_{m i, j, s - l} (t) ~ n_{m} [1 \pm \frac{1}{2} N (μ (t) = Δ n_{m, j}, σ^{2} = δ n_{m, l})] .$

3.2 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 <Δn_m,j> is indicated in Table 1.

Fig. 4 Schematic cross-sections of multi-core fibers simulated: (a) lowly- and (b) highly-birefringent cores.

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Table 1. Number of segments and linear birefringence average values considered in the analysis

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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 <Δn_m,j> is assumed in a given core m, the lower is the number of segments N_m that should be considered. Additional MCF parameters employed were: d_ab = 36 μm, d_c = 125 μm, R₀ = 4 μm, n_a = n_b = 1.45, n_c = 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 <Δn_m,j> in each core m = a,b. In HB-MCFs with elliptical cores, the intrinsic birefringence can also be included in <Δn_m,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 Δn_a,j and Δn_b,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 μ_υ,ξ: = E[P_υ(z = L)/P_ξ(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 R_B = 100 cm along the MCF. In addition, two different twist rate average values of f_T = 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 Δn_m,j (m = a,b) depicted in Figs. 5(a1) and 5(a2) was calculated from a Gaussian distribution N(<Δn_m,j>,10⁻⁷) considering a similar evolution in both cores in coherence with the experimental results of the next section.

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 f_T = 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).

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As we can notice from Figs. 5(a1) and 5(a2), the higher the linear birefringence Δn_m,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 LP_01,_mx and LP_01,_my. If the linear birefringence Δn_m,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 1st, 3rd,5th, 6th and 8th days. As a result, the XIC-XT mean is higher than DIC-XT these days, as shown in Fig. 5(c2).

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 <Δn_m,j> = 4·10⁻⁴ in both cores (m = a,b), as indicated in Table 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. Figures 6(a1)-6(c1) and Figs. 6(a2)-6(c2) show the simulated results for twist rate average values of f_T = 0.05 and 1 turns/m, respectively.

Fig. 6 Simulation results of the crosstalk behavior between the polarized core modes in a 150 m two-core fiber with highly-birefringent cores considering the temporal fluctuation of the linear birefringence of each core with a twist rate value of f_T = 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).

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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 Δn_m,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 <Δn_m,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 f_T 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 R_B, for which the condition Δϕ_ξυ ≈Δϕ_υξ 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 <Δn_aj> = <Δn_bj> = 10⁻⁷. The temporal birefringence fluctuation of the TCF was omitted in this simulation. The numerical results are shown in Fig. 7.

Fig. 7 Multi-parameter simulation of the crosstalk between polarized core modes varying the bending radius (1/R_B) and the twist rate (f_T) 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.

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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 R_B and fiber twisting conditions f_T. As it can be noticed from Fig. 7(a), we cannot observe intra-core mode coupling between ax-ay with f_T = 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 f_T = 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, f_T = 0.08, R_B = 100 cm, N_m = 2 segments, S = 80 iterations, d_ab = 36 μm, d_c = 125 μm, R₀ = 4 μm, n_a = n_b = 1.45, n_c = 1.44, and λ = 1550 nm, with a constant birefringence value in each core of Δn_a₍_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.

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).

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It should be noted from Fig. 8(a) that the iC-XT mean remains unchanged for a power launch level P_L = P_ax(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.

4. 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.

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.

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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. An optical 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.

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 (LP_01,_mx and LP_01,_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 <Δn_1,_j> = 4.9·10⁻⁷ and <Δn_3,_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.

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 P_L = 0 dBm was used in linear regime (NL: nonlinear regime with P_L = 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

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As depicted in Figs. 10(a) and 10(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 3rd and 4th days (October 2015), the 7th day (January 2016), and the 13th and 14th 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 = N_a = 8, N₃ = N_b = 6 and assuming a temporal birefringence Δn₁_,j = Δn_a,j and Δn₃_,j = Δn_b,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 5th, 6th and 15th 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 Δn_m,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 <Δn_m,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 <Δn_m,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): n_m = 1.452, n_c = 1.444, λ = 1550 nm, d_ab = 36 μm, d_c = 125 μm. The distance to the reference center of the 4CF is fixed at d_m = d_ab/√2 = 25.46 μm. The bending radius and the twist rate was assumed with a similar value as in the experimental set-up with R_B = 100 cm and f_T = 1 turns/m, respectively. It can be noted that the iC-XT mean is reduced when the average value of the linear birefringence <Δn_m,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.

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).

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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.

5. 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 m_ax,ay (z;t) and η_ax,by (z;t) were found modeling iC-XT and XIC-XT, respectively. The theoretical analysis shows that η_ax,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 n_mi^(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 DIC- and 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 F_mi(x,y;z,t)

The transversal local eigenfunction F_mi 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 n_mi^(eq) detailed in Section 3. Therefore, the eigenfunction F_mi(r;z,t) can be expressed in a local polar coordinate system (r,φ,z) of a given core m as:

F_{m i} (r; z, t) = {\begin{matrix} J_{0} (\frac{u_{m i} (z; t)}{R_{0}} r); r \leq R_{0} \\ K_{0} (\frac{w_{m i} (z; t)}{R_{0}} r); r > R_{0} \end{matrix},

where r = (x² + y²)^1/2; R₀ is the core radius (assuming all cores of the MCF with identical radius); u_mi and w_mi are the modal parameters of the PCM “mi”, which can be calculated as:

u_{m i} (z; t) = (1 + \sqrt{2}) V_{m i} (z; t) / [1 + {(4 + V_{m i}^{4} (z; t))}^{1 / 4}],

w_{m i} (z; t) = \sqrt{V_{m i}^{2} (z; t) - u_{m i}^{2} (z; t)},

and V_mi(z;t) is the normalized frequency of the PCM “mi”:

V_{m i} (z; t) = k_{0} R_{0} \sqrt{{(n_{m i}^{(eq)} (z; t))}^{2} - n_{c}^{2}},

with n_mi^(eq) given by Eq. (19) and n_c the material refractive index of the cladding. Note that in Eq. (24) it was assumed that the material refractive index of the cladding is similar for the x and y polarization axes.

Funding

This work has been partly funded by Spain National Plan project MINECO/FEDER UE XCORE TEC2015-70858-C2-1-R; HIDRASENSE RTC-2014-2232-3; European Regional Development Fund (ERDF) and the Galician Regional Government under project GRC2015/018. 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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Birefringence effects in multi-core fiber: coupled local-mode theory

Abstract

1. Introduction