Analytical expressions for the crosstalk of super-modes in the tightly bounded multicore fibers

Junhe Zhou; Junhe Zhou; Haoqian Pu

doi:10.1364/OE.448674

1. Introduction

Multicore fibers (MCFs) have been viewed as one of the promising approaches to further increase the optical fiber system transmission capacity [1–4]. MCF based transmission systems can be either digital signal processing (DSP) based or non-DSP based. While the former compensates the inter-core/mode crosstalk via the DSP chips with the design aim to reduce the delay spread [5–10], the latter aims to minimize the crosstalk between the transmission modes.

There are two main design categories for the crosstalk minimized MCFs. One targets to minimize the inter-core crosstalk, which adopts the core trench, a large core index difference, and a relatively long core pitch (the distance between the core centers) between the neighboring cores [11–22]. The other design aims to use the super-modes for transmission and to minimize the inter-super-mode crosstalk (ISMXT) by increasing the coupling strength so that the super-modes have more significant propagation constant difference. Therefore, it has a much shorter core pitch [23–26]. Sometimes, the core pitch is twice of the core radius, i.e., the two neighboring cores have zero spacing [23]. In comparison to the former, the latter has a closer core arrangement and therefore the MCFs can contain more cores/transmission modes within the finite cross section.

Crosstalk analysis has been a very important topic in the MCF design. The random inter-core crosstalk [11–22] has been studied quite intensively through the coupled mode equations (CMEs) and various models have been proposed to address the problem more accurately and efficiently. In [22], the polarization mode dispersion (PMD) was incorporated in the coupling equation and it was demonstrated that the PMD tended to reduce the crosstalk in case the birefringence correlation length and the coupling correlation length were comparable.

Despite the pioneering work in [11–22], there have been much fewer theoretical studies on the random crosstalk among the super-modes. S. Saitoh et. al. [23] demonstrated the experimental measurements for the crosstalk among the super-modes in a four-core fiber. C. Xia et. al. designed a three-core fiber with the numerical simulations [24–25] and discussed the idea to have a larger effective index difference for the super-modes to reduce the crosstalk. However, no quantitative theoretical predictions for the super-mode crosstalk were provided in [24–25]. In [26], the deterministic crosstalk among the OAM super-modes was discussed but the random crosstalk was not considered. The lack of analytical study is caused by the fact that the super-mode transmission is quite different from the standard MCF transmission in the individual cores [23–27]. It resembles the multimode transmission in multimode fibers (MMFs). Therefore, the standard approach for the crosstalk evaluation between the cores [11,22] cannot be applied directly for the super-mode crosstalk evaluation. The main difficulty to derive a comprehensive theory for the analytical study on the random crosstalk among the super-modes is the complicated interactions between the deterministic coupling effects and the random coupling effects. Their coupling beat length and correlation length are comparable and it makes the derivation extremely difficult.

In our latest work on the topic of crosstalk in MCFs [28], a unitary transformation method is proposed to study the random crosstalk inside MCFs with deterministic coupling. However, the MCFs under investigation in [28] had their deterministic coupling beat length to be comparable with the PMD correlation length, which suggests the deterministic coupling strength is not strong enough to hold the stable super-mode. In addition, no detailed discussions on the explicit expressions for the super-mode crosstalk were provided in [28].

In this work, we provide a detailed analytical framework for the super-mode random crosstalk in the tightly bounded MCFs. The theory is based on the stochastic differential equation (SDE). The deterministic coupling among the cores is eliminated through the unitary transform, which also re-distributes the inter-core random coupling effects among the super-modes. When the deterministic coupling strength is strong (which is true in MCFs designed for super-modes transmission), the deterministic coupling beat length will be much shorter than the birefringence correlation length, and the impact of polarization effect on the ISMXT can be neglected. The updated SDE for the super-modes is used to derive the coupled ordinary differential equations (ODEs) for the super-mode power, as well as for the super-mode power variance and the power correlations. The latter can be useful for the system crosstalk budget estimation and the system capacity analysis. Explicit and simple formulas are given for the characterization of the crosstalk among the super-modes under the weak ISMXT assumption.

2. Theory

In the tightly bounded MCFs designed for super-mode transmission, the deterministic coupling beat length (in the order of mm [23]) is much shorter than the PMD correlation length (in the order of meters [22]). In this case, the impact of PMD on the super-mode crosstalk can be neglected, because the polarization of the signal does not change within the coupling beat length. (The Monte Carlo simulation will consider the PMD impact and the results will be compared with the analytical results derived here). Therefore, the scalar coupled mode theory (CMT) can be used to characterize the propagation of the amplitudes in the cores:

(1)$$\begin{aligned} &\frac{{d\textbf{E}}}{{dz}} = i\textbf{BE} + i\boldsymbol{\mathrm{\kappa}} \circ \textbf{f}(z )\textbf{E}\\ &\textbf{B} = \left( {\begin{array}{cccc} {{\beta_1}}&{{\beta_{12}}}&{}&{{\beta_{1N}}}\\ {}&{{\beta_2}}&{}&{}\\ {}&{}& \ddots &{}\\ {{\beta_{N1}}}&{}&{{\beta_{NN - 1}}}&{{\beta_N}} \end{array}} \right),\\ &\boldsymbol{\kappa } \circ \textbf{f}(z )= \left( {\begin{array}{cccc} {{\kappa_{11}}{f_{11}}(z )}& \cdots &{}&{{\kappa_{1N}}{f_{1N}}(z )}\\ \vdots &{{\kappa_{22}}{f_{22}}(z )}&{}& \vdots \\ {}&{}& \ddots &{}\\ {{\kappa_{N1}}{f_{N1}}(z )}& \cdots &{}&{{\kappa_{NN}}{f_{NN}}(z )} \end{array}} \right), \end{aligned}$$

where E is a vector containing the element E_m as the amplitude of the mth core, and B is the deterministic coupling matrix with its diagonal element β_m as the propagation constant of the mth core, which includes the chromatic dispersion, and its non-diagonal element β_mn as the deterministic coupling coefficient between the mth core and the nth core. The random coupling part is contained in the matrix $\boldsymbol{\kappa } \circ \textbf{f}(z )$, where κ_mn and f_mn stand for the random coupling amplitude and the z directional random variation process between the mth and the nth cores. The deterministic part and the random part of the coupling represent the intrinsic coupling effects among the cores and the random coupling variations caused by the perturbations, such as fiber bending and core misalignment. The random variation process f_mn is a filtered color noise with the correlation function as [12–13,21–22]

(2)$$\left\langle {{f_{mn}}({z + z^{\prime}} ){f_{mn}}^\ast ({z^{\prime}} )} \right\rangle = {\eta _{mn}}(z ).$$

While it is not mandatory, it is often assumed that f_mn are identically distributed independent (i.i.d) variables, and the correlation function ${\eta _{mn}}$ is assumed as the two-side exponential function [12–13,21–22]:

(3)$${\eta _{mn}}(z )= \eta (z )= \exp \left( { - \frac{{|z |}}{{{l_c}}}} \right).$$

where l_c is the random coupling correlation length. The Fourier transform of η is

(4)$$\textbf{FT}(\eta )(\omega )= \frac{{2{l_c}}}{{1 + {l_c}^2{\omega ^2}}},$$

where FT stands for the Fourier transform, and ω is the angular frequency.

When B is a diagonal matrix, Eq. (1) has been well solved [12–13,21]. When it is not diagonal, it can be dealt with as follows. Since B is a Hermitian matrix, it can be decomposed as HDH^H, where D is a diagonal matrix and H is a unitary matrix [28]. Make the following substitution, it can convert Eq. (1) to Eq. (5) [28].

(5)$$\begin{aligned} &\textbf{E}^{\prime} = {\textbf{H}^H}\textbf{E},\\ &\boldsymbol{\mathrm{\kappa}}^{\prime}(z )= {\textbf{H}^H}\boldsymbol{\kappa } \circ \textbf{f}(z )\textbf{H},\\ &\frac{{d\textbf{E}^{\prime}}}{{dz}} = i\textbf{DE}^{\prime} + i\boldsymbol{\mathrm{\kappa}}^{\prime}(z )\textbf{E}^{\prime}. \end{aligned}$$

The correlation function for the element of matrix $\boldsymbol{\mathrm{\kappa}}^{\prime}(z )$ is

(6)$$\left\langle {{{\kappa^{\prime}}_{mn}}({z + z^{\prime}} ){{\kappa^{\prime}}_{mn}}^\ast ({z^{\prime}} )} \right\rangle = \left( {\sum\limits_{k = 1}^N {\sum\limits_{l = 1}^N {{{|{{H_{km}}} |}^2}{{|{{H_{ln}}} |}^2}{{|{{\kappa_{kl}}} |}^2}} } } \right)\eta (z ).$$

In MCFs designed for the super-mode transmission, the super-modes are non-degenerated and they have different propagation constants, i.e., the diagonal matrix D has different diagonal elements (d₁,..d_N). By substituting $\textbf{A} = \exp ({ - i\textbf{D}z} )\textbf{E}^{\prime}$, Eq. (5) can be rewritten as [12]

(7)$$\frac{{d\textbf{A}}}{{dz}} = \exp ({ - i\textbf{D}z} )i\boldsymbol{\mathrm{\kappa}}^{\prime}(z )\exp ({i\textbf{D}z} )\textbf{A}.$$

Define matrix F as

(8)$$\textbf{F} = \exp ({ - i\textbf{D}z} )i\boldsymbol{\mathrm{\kappa}}^{\prime}(z )\exp ({i\textbf{D}z} ).$$

In case the random coupling correlation length l_c is much shorter than the propagation length, the element of F has following property:

(9)$$\begin{aligned} &\left\langle {{F_{mn}}dz{F_{mn}}^\ast dz} \right\rangle = {K_{mn}}dz,\\ &\left\langle {{F_{mn}}dz{F_{m^{\prime}n^{\prime}}}^\ast dz} \right\rangle = 0({m \ne m^{\prime},n \ne n^{\prime}} ),\\ &{K_{mn}} = \left( {\sum\limits_{k = 1}^N {\sum\limits_{l = 1}^N {{{|{{H_{km}}} |}^2}{{|{{H_{ln}}} |}^2}{{|{{\kappa_{kl}}} |}^2}} } } \right)\textbf{FT}(\eta )({{d_m} - {d_n}} ), \end{aligned}$$

Equation (7) is the Stratonovich sense SDE, and it should be converted to the Ito sense SDE [28]:

(10)$$d\textbf{A} = i\textbf{F}dz\textbf{A} - \frac{1}{2}\left( {\begin{array}{ccc} {\sum\limits_{m = 1}^N {{K_{1m}}} }&{\cdots 0 \cdots}&0\\ \vdots & \ddots & \vdots \\ 0&{ \cdots 0 \cdots }&{\sum\limits_{m = 1}^N {{K_{Nm}}} } \end{array}} \right)dz\textbf{A}.$$

Defining the power of the mth super-mode as < P_m>=<e’_me’_m*>=<a_ma_m*>, with e_m’ and a_m being the elements of the vectors E’ and A, one may derive the coupled power equation based on Eqs. (9–10) [28–30]:

(11)$$\frac{{d\left\langle {{P_m}} \right\rangle }}{{dz}} = \sum\limits_{m^{\prime} = 1}^N {{K_{mm^{\prime}}}\left( {\left\langle {{P_{m^{\prime}}}} \right\rangle - \left\langle {{P_m}} \right\rangle } \right)} ,$$

and the coupled power correlation equation can also be derived [28,30]:

(12a)$$\frac{{d\left\langle {{P_m}^2} \right\rangle }}{{dz}} ={-} 2\sum\limits_{m^{\prime} = 1}^N {{K_{mm^{\prime}}}} \left\langle {{P_m}^2} \right\rangle + 4\sum\limits_{m^{\prime} = 1}^N {{K_{mm^{\prime}}}} \left\langle {{P_m}{P_{m^{\prime}}}} \right\rangle - 2\left\langle {{P_m}{{(z )}^2}} \right\rangle {K_{mm}},$$

(12b)$$\frac{{d\left\langle {{P_m}{P_n}} \right\rangle }}{{dz}} = \left( { - \sum\limits_{m^{\prime} = 1}^N {{K_{mm^{\prime}}}} - \sum\limits_{n^{\prime} = 1}^N {{K_{nn^{\prime}}}} } \right)\left\langle {{P_m}{P_n}} \right\rangle + \sum\limits_{m^{\prime} = 1}^N {{K_{mm^{\prime}}}\left\langle {{P_n}{P_{m^{\prime}}}} \right\rangle } + \sum\limits_{n^{\prime} = 1}^N {{K_{nn^{\prime}}}\left\langle {{P_m}{P_{n^{\prime}}}} \right\rangle } - 2{K_{mn}}\left\langle {{P_m}{P_n}} \right\rangle .$$

Since Eqs. (11–12) are the ODEs with constant coefficients, they can be solved analytically [28,30].

We may use the above theory to analyze a two-core MCF. It has a symmetrical super-mode (super-mode 1) and an anti-symmetrical super-mode (super-mode 2). Furthermore, it can be noticed that inter-super-mode coupling coefficients K₁₂= K₂₁. Hence, Eq. (11) can be solved as

(13)$$\begin{aligned} \left\langle {{P_1}(z )} \right\rangle &= \frac{1}{2}({1 + \exp ({ - 2{K_{12}}z} )} )\left\langle {{P_1}(0 )} \right\rangle + \frac{1}{2}({1 - \exp ({ - 2{K_{12}}z} )} )\left\langle {{P_2}(0 )} \right\rangle \\ \left\langle {{P_2}(z )} \right\rangle &= \frac{1}{2}({1 + \exp ({ - 2{K_{12}}z} )} )\left\langle {{P_2}(0 )} \right\rangle + \frac{1}{2}({1 - \exp ({ - 2{K_{12}}z} )} )\left\langle {{P_1}(0 )} \right\rangle \end{aligned}$$

The ISMXT from super-mode 2 to super-mode 1 can be evaluated as

(14)$$ISMXT = \frac{1}{2}\frac{{({1 - \exp ({ - 2{K_{12}}z} )} )\left\langle {{P_2}(0 )} \right\rangle }}{{\left\langle {{P_2}(0 )} \right\rangle }} = \frac{1}{2}({1 - \exp ({ - 2{K_{12}}z} )} )\approx {K_{12}}z({{K_{12}}z \ll 1} )$$

In the case of weak crosstalk, i.e., K₁₂z<<1, we have ISMXT to be approximately K₁₂z, which resembles the inter-core crosstalk analysis, albeit with the change of the definition for the crosstalk coefficient K₁₂ by Eq. (6). Similarly, Eq. (12) can be solved as

(15)$$\begin{aligned} &\left\langle {{P_1}{{(z )}^2}} \right\rangle = \left( {\frac{1}{6}\exp ({ - 6{K_{12}}z} )+ \frac{1}{2}\exp ({ - 2{K_{12}}z} )+ \frac{1}{3}} \right)\left\langle {{P_1}{{(0 )}^2}} \right\rangle + \left( { - \frac{2}{3}\exp ({ - 6{K_{12}}z} )+ \frac{2}{3}} \right)\left\langle {{P_1}(0 ){P_2}(0 )} \right\rangle \\ &+ \left( {\frac{1}{6}\exp ({ - 6{K_{12}}z} )- \frac{1}{2}\exp ({ - 2{K_{12}}z} )+ \frac{1}{3}} \right)\left\langle {{P_2}{{(0 )}^2}} \right\rangle \\ &\left\langle {{P_1}(z ){P_2}(z )} \right\rangle = \left( {\frac{1}{6} - \frac{1}{6}\exp ({ - 6{K_{12}}z} )} \right)\left\langle {{P_1}{{(0 )}^2}} \right\rangle + \left( {\frac{2}{3}\exp ({ - 6{K_{12}}z} )+ \frac{1}{3}} \right)\left\langle {{P_1}(0 ){P_2}(0 )} \right\rangle \\ &+ \left( {\frac{1}{6} - \frac{1}{6}\exp ({ - 6{K_{12}}z} )} \right)\left\langle {{P_2}{{(0 )}^2}} \right\rangle \\ &\left\langle {{P_2}{{(z )}^2}} \right\rangle = \left( {\frac{1}{6}\exp ({ - 6{K_{12}}z} )+ \frac{1}{2}\exp ({ - 2{K_{12}}z} )+ \frac{1}{3}} \right)\left\langle {{P_2}{{(0 )}^2}} \right\rangle + \left( { - \frac{2}{3}\exp ({ - 6{K_{12}}z} )+ \frac{2}{3}} \right)\left\langle {{P_1}(0 ){P_2}(0 )} \right\rangle \\ &+ \left( {\frac{1}{6}\exp ({ - 6{K_{12}}z} )- \frac{1}{2}\exp ({ - 2{K_{12}}z} )+ \frac{1}{3}} \right)\left\langle {{P_1}{{(0 )}^2}} \right\rangle \end{aligned}$$

Hence, the variance of ISMXT from super-mode 2 to super-mode 1 is

(16)$$\begin{aligned} &{\mathop{\textrm{var}}} ({ISMXT} )= \left( {\frac{1}{6}\exp ({ - 6{K_{12}}z} )- \frac{1}{2}\exp ({ - 2{K_{12}}z} )+ \frac{1}{3}} \right) - {\left( {\frac{1}{2}({1 - \exp ({ - 2{K_{12}}z} )} )} \right)^2}\\ &= \frac{1}{{12}} - \frac{1}{4}\exp ({ - 4{K_{12}}z} )+ \frac{1}{6}\exp ({ - 6{K_{12}}z} )\approx {({{K_{12}}z} )^2}({{K_{12}}z \ll 1} )\end{aligned}$$

In the case of weak crosstalk, i.e., K₁₂z<<1, we have the variance of ISMXT from super-mode 2 to super-mode 1 to be approximately (K₁₂z)². Therefore, the standard deviation should be K₁₂z. The results in Eqs. (9,14,16) can be used to evaluate the ISMXT power and power standard deviation between the mth super-mode and the nth super-mode by replacing the inter-super-mode coupling coefficient K₁₂ by K_mn in case the crosstalk is weak.

As shown above, the power coupling coefficient K₁₂ which describes the random crosstalk strength between the super-modes is very important for ISMXT characterization. It can be calculated as

(17)$${K_{12}} = \frac{{2{l_c}}}{{1 + {l_c}^2{{({2{\beta_{12}}} )}^2}}}\left( {\frac{1}{4}{{|{{\kappa_{12}}} |}^2} + \frac{1}{4}{{|{{\kappa_{21}}} |}^2}} \right)$$

Considering the random coupling amplitude is r times proportional to the deterministic part, i.e., $|{{\kappa_{12}}} |= |{{\kappa_{21}}} |= r{\beta _{12}}$, we have the limit for K₁₂ in case of strong deterministic coupling, which should be proportional to the square of the ratio r and inversely proportional to the random coupling correlation length l_c.

(18)$$\mathop {\lim }\limits_{{\beta _{12}} \to \infty } {K_{12}} = \frac{{{r^2}}}{{4{l_c}}}$$

3. Results and discussions

Monte Carlo simulations have been conducted to verify the proposed analytical model. A two-core fiber discussed above along with a four-core MCF fabricated in [23] are discussed in detail. The advantage of such a four-core fiber is that it has four non-degenerated super-modes which can be used for MIMO-less transmission. The two-fiber and the four-core fiber cross sections are shown in Fig. 1.

Fig. 1. The cross section of (a) the two-core fiber mentioned above (b) The four-core fiber mentioned in [23].

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The fiber parameters are the same as those in [23]. The step index fiber has the cladding index as 1.45, and the core index difference as 0.31% with respect to the cladding index. The core radius is 2.5 µm and the cladding diameter is 125.7 µm. The core pitch for the two-core fiber is $5\sqrt 3 =$ 8.66 µm. There are two core pitches in the four-core MCF to create the non-degenerated super-modes, which are 5µm and $5\sqrt 3 =$ 8.66µm respectively. The signal wavelength is 1.55µm. To facilitate the readers, the detailed parameters for the four-core fiber in [23] is shown in Table 1.

Table 1. Detailed Parameters of the 4-Core Fiber in [23]

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For the two-core fiber, the deterministic coupling coefficients between the two cores can be calculated as β₁₂= 960/m. For the four-core fiber, the deterministic coupling coefficients between core 1 and core 2, between core 1 and core 3, and between core 1 and core 4 can be calculated as β₁₂= 3031/m, β₁₃= 1285/m and β₁₄= 960/m respectively. These evaluated values are in accordance with the reported values in [23]. While there are two super-modes in the two-core fiber, there are four super-modes in the four-core MCF with the following propagation constants [23].

(19)$$\begin{aligned} &{\beta _{LP01 - like}} = {\beta _0} + {\beta _{12}} + {\beta _{13}} + {\beta _{14}}\\ &{\beta _{LP11a - like}} = {\beta _0} + {\beta _{12}} - {\beta _{13}} - {\beta _{14}}\\ &{\beta _{LP11b - like}} = {\beta _0} - {\beta _{12}} + {\beta _{13}} - {\beta _{14}}\\ &{\beta _{LP21 - like}} = {\beta _0} - {\beta _{12}} - {\beta _{13}} + {\beta _{14}} \end{aligned}$$

where β₀ is the common propagation constant of each of the four cores. The four super-modes in Eq. (19) can be denoted as super-mode 1-4 sequentially, and the inter super-mode coupling coefficients can be denoted as K₂₁ K₃₁, K₄₁ …etc.

In the Monte Carlo simulations, the full vector model is used [22].

(20)$$\frac{{d{{\vec{E}}_m}}}{{dz}} = i{\beta _m}{\vec{E}_m} + \sum\limits_{n = 1}^{N - 1} {i{\beta _{mn}}{{\vec{E}}_n}} + \sum\limits_{n = 1}^N {i{\kappa _{mn}}{f_{mn}}(z ){{\vec{E}}_n}} + \frac{{i{{\vec{\alpha }}_m}}}{2}\cdot \vec{\sigma }{\vec{E}_m},$$

where vector ${\vec{E}_m}$ is the Jones vector of the mth core, $\vec{\sigma }$ is the vector whose elements are the Pauli matrices, ${\vec{\alpha }_m}$ is the random birefringence vector of the mth core, and its correlation function fulfills [22]

(21)$$\left\langle {{{\vec{\alpha }}_n}({z + z^{\prime}} )\cdot {{\vec{\alpha }}_n}({z^{\prime}} )} \right\rangle = {\left( {\frac{{2\pi }}{{{L_B}}}} \right)^2}\exp \left( { - \frac{{|z |}}{{{L_C}}}} \right),$$

where L_C is the birefringence correlation length, which describes how the birefringence autocorrelation function attenuates, and L_B is the birefringence beat length, which describes the birefringence strength.

L_C is assumed to be 1 m and L_B is assumed to be 10 m. These random parameters are used and discussed in the published papers [22]. The random variation of the coupling coefficients is assumed to have the root mean square (rms.) value of 0.002β₁₂ for the two-core fiber and 0.002β₁₂, 0.002β₁₃, 0.002β₁₄ respectively for the four-core fiber. To obtain the average value for the power and the power correlation terms, 1000 realizations are evaluated in the Monte Carlo simulations.

Since the birefringence correlation length is much shorter than the beating length for the super-modes deterministic coupling (in the order of mm [23]), our proposed scalar analytical model, which ignores the polarization effect on the crosstalk, i.e., Eqs. (11–12) can be used.

The two-core fiber is simulated at first. The random coupling correlation length l_c is directly related to the crosstalk between the super-modes as shown in section 2, and it is assumed to be 1 m. It is assumed that the unit power is injected on the symmetrical super-mode, and its crosstalk to the anti-symmetrical super-mode is calculated in Figs. 2(a-b). The dash dotted line, the dashed line and the solid line represent the results by the Monte Carlo simulations, the simple formulas by Eqs. (9,14,16), and the analytical model. It can be seen from the figure that the average crosstalk power and the power standard deviation are accurately predicted by the analytical model and the simple formula in comparison to the Monte Carlo simulations. It can also be inferred from the figure that the crosstalk level can vary by 3 dB. Such information can be very useful during the system design and analysis, especially during the consideration of the receiver signal to noise ratio (SNR) budget.

Fig. 2. The evolution of the crosstalk power and power standard deviation from the symmetrical super-mode to the anti-symmetrical super-mode. The random coupling correlation length l_c= 1 m. The unit power is injected on the symmetrical super-mode. The results are computed by the Monte Carlo simulations, the explicit formulas in Eqs. (9,14,16), and the analytical model.

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Afterwards, we check the impact of l_c on the crosstalk. Its value varies as 1m, 4m, 7m and 10m respectively. The fiber length is 100m. The curves in Figs. 3(a-b) suggest that the average crosstalk power and the power standard deviation are accurately predicted by the analytical model compared with the Monte Carlo simulations, which demonstrates the robustness of the model. Also, it can be concluded from the figure that the average crosstalk power and the power standard deviation are inversely proportional to l_c, which is shown in Eq. (18). The discrepancy between the curves is caused by the following factor. The theory is derived based on the assumption that the random coupling correlation length is much shorter than the propagation distance, which allows the random process to be treated as the Brownian motion. When the propagation distance is shorter or comparable to the random coupling correlation length, deviation from the theory appears, which is the case at the beginning of the propagation. Quantitative investigations on the discrepancy are shown in Figs. 3(c-d). As expected, the discrepancy grows as l_c increases. When the propagation distance increases, the discrepancy will reduce accordingly.

Fig. 3. The change of the crosstalk power and the power standard deviation from the symmetrical super-mode to the anti-symmetrical super-mode with respect to the value of l_c. The unit power is injected on the symmetrical super-mode and the fiber length is 100 m. The results are computed by the Monte Carlo simulations and the analytical model. The discrepancy between the two models is shown in (c) and (d).

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Thereafter, the four-core fiber is simulated. The random coupling correlation length l_c is also assumed to be 1 m at first. It is assumed that the unit power is injected on the LP21-like super-mode, and the crosstalk to other super-modes are evaluated accordingly. The results are shown in Figs. 4(a-c). It can be concluded that the proposed analytical model and the simple formulas, i.e., Eqs. (9,14) give very satisfactory predictions as compared with the results by the Monte Carlo simulations. The crosstalk between LP21-like mode and the LP11b-like mode is much higher than the other two super-modes. This is because the propagation constants difference is much less between the LP21-like mode and the LP11b-like mode. The evolution of the standard deviation for the crosstalk between the LP21-like mode and the LP11b-like mode is shown in Fig. 4(d). It can be seen from the figure that the proposed analytical model and the simple formulas Eqs. (9,16) do have a very high accuracy when evaluating the power variation of the super-mode crosstalk.

Fig. 4. The evolution of the crosstalk power from the LP21-like mode to the (a) LP11b-like mode, (b) LP11a-like mode, and (c) LP01-like mode. (d) The evolution of the crosstalk power standard deviation from the LP21-like mode to the LP11b-like mode. The random coupling correlation length l_c= 1 m. The unit power is injected on LP21-like mode. The results are computed by the Monte Carlo simulations, the explicit formulas in Eqs. (9,14,16), and the analytical model.

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It should be noted that the simple formulas, i.e., Eqs. (9,14,16) deviate from the analytical results when the crosstalk level is high, e.g., −10 dB. In this way, the analytical solution will be a more accurate approach to predict the ISMXT power and power standard deviation.

Afterwards, the correlation length l_c is assumed to be 10m. Still, we assumed that the unit power is injected on the LP21-like super-mode. The results for the crosstalk are shown in Figs. 5(a-c). The agreement between the Monte Carlo simulations, the simple formulas Eqs. (9,14), and the analytical model demonstrates the validity of the model once more. Comparing with the results in Figs. 4(a-c), it is clearly shown that the crosstalk level, which is inversely proportional to the random correlation length l_c, has been reduced by a factor of 10. One may infer that manufacturing the MCFs with high consistency (i.e., longer correlation length for the variation) will reduce the crosstalk between the super-modes. The final crosstalk level between the LP21-like mode and the LP11b-like mode in Fig. 5(a) is about −23dB/km, which is the same as the measurement results in [23], suggesting the parameter setup might be close to the actual parameters in the fabrication [23].

Fig. 5. The evolution of the crosstalk power from the LP21-like mode to the (a) LP11b-like mode, (b) LP11a-like mode, and (c) LP01-like mode. (d) The evolution of the crosstalk power standard deviation from the LP21-like mode to the LP11b-like mode. The random coupling correlation length l_c= 10 m. The unit power is injected on LP21-like mode. The results are computed by the Monte Carlo simulations, the explicit formulas in Eqs. (9,14,16), and the analytical model.

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The evolution of the standard deviation for the crosstalk between the LP21-like mode and the LP11b-like mode when l_c= 10m is shown in Fig. 5(d). The consistency between the three lines further confirms the validity of the proposed model. When the crosstalk level decreases by 10 time as l_c increases, the standard deviation for the crosstalk also decreases accordingly.

It is worth noting that the presented theory can be applied to any kind of MCFs designed for super-mode transmission with an arbitrary core arrangement. The specifically designed rectangular 4-core fiber presented above is used to realize MIMO-less transmission, with the super-modes being non-degenerated. As long as the super-modes are non-degenerated, the theory can produce accurate predictions for the super-mode crosstalk. In the standard non-rectangular MCF, four cores might be arranged as a square, within which the degenerated super-modes will appear. The theory is still able to predict the crosstalk between the super-modes with different propagation constants, but it is not able to predict the crosstalk between the super-modes with the same propagation constants. Fortunately, the crosstalk between the super-modes with different propagation constants is of more concern [24].

4. Summary

An analytical approach for evaluating the super-mode crosstalk in the tightly bounded MCFs is proposed and its accuracy is tested with the Monte Carlo simulations. The derived results suggest that the crosstalk level can be reduced if the propagation constant difference is large and the correlation length of the random variation is long. When the deterministic coupling strength is strong, it will result in higher propagation constant difference and thus a much shorter coupling beat length can be expected. This suggests the polarization of the signal barely changes during the coupling process and it will suppress the impact of PMD. If the MCF is manufactured with high consistency and the random variation correlation length is long, the crosstalk level can also be reduced. The theoretical results presented in this work can be useful for the MCF design and analysis.

Funding

National Natural Science Foundation of China (61775168).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Parameters	Value
Number of cores	4
Core radius	2.5µm
Cladding diameter	125.7µm
Core pitch 1	5µm
Core pitch 2	8.66µm
cladding index	1.45
Core index difference	0.31%

Analytical expressions for the crosstalk of super-modes in the tightly bounded multicore fibers

Abstract

1. Introduction

2. Theory

3. Results and discussions

4. Summary

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Tables (1)

Equations (22)

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