Simple analytical expression for crosstalk estimation in homogeneous trench-assisted multi-core fibers

: An analytical expression for the mode coupling coe ﬃ cient in homogeneous trench-assisted multi-core ﬁbers is derived, which has a simple relationship with the one in normal step-index structures. The amount of inter-core crosstalk reduction (in dB) with trench-assisted structures compared to the one with normal step-index structures can then be written by a simple expression. Comparison with numerical simulations conﬁrms that the obtained analytical expression has very good accuracy for crosstalk estimation. The crosstalk properties in trench-assisted multi-core ﬁbers, such as crosstalk dependence on core pitch and wavelength-dependent crosstalk, can be obtained by this simple analytical expression.


Introduction
Space-division multiplexing (SDM) is a promising candidate to overcome the capacity limit of single-mode fibers (SMFs) so as to sustain the rapid Internet traffic growth [1]. SDM based on uncoupled multi-core fibers (MCFs) is a simple and robust option, which requires no complex multiple-input multiple-output signal processing at the receiver side.
The main issue with high capacity MCF transmission is how we can increase the number of cores (thus, total capacity) while keeping the inter-core crosstalk (XT) low to allow higher multi-level modulation formats and longer transmission distance since more dense cores (smaller core pitch) normally increase the modes coupling among cores. The trench-assisted structure, based on its strong mode confinement capability, was originally proposed to reduce the fiber bending loss for FTTH applications [2]. The structure was then proposed to be applied to each core in MCFs for XT reduction [3], and capacity records experiments of both 305 Tb/s [4] and 1.01 Pb/s [5] transmission adopted trench-assisted structures, taking advantage of its low XT. Other XT reduction techniques such as hole-assisted [6], hole-walled [7] structures were also proposed. The relationships between the XT and the fiber parameters, such as relative refractive index difference between trench and cladding, trench width in trench-assisted multi-core fibers (TA-MCFs), however, have not been formulated analytically, and thus, the estimations of the XT in TA-MCFs in all the reported works so far [3, 8-10] have been done by numerical simulations and experimental measurements. In order to design MCFs with ultra-low XT for long distance and high capacity MCF transmission, however, it is important to understand the effects of these various fiber parameters on the XT reduction in TA-MCFs based on an analytical expression.
In this paper, we derive an analytical expression for the mode coupling coefficient in TA-MCFs, where the amount of XT reduction (in dB) by trench-assisted structure relative to normal step-index structure is given by a simple expression. The XT calculated by the derived analytical expression agree very well with those simulated by finite element method. In addition to the possibility of fast and accurate XT estimations in trench-assisted structure, the analytical expression can also be used to analyze the XT properties, such as XT dependence on core pitch [11] and wavelength-dependent XT [12]. This paper is divided into 5 sections. In section 2, the mode coupling coefficient in trenchassisted structure is formulated by relating to the one in normal step-index structure. Some appropriate approximation techniques are used in the derivation so that it can be simplified into an analytical expression. In section 3, the derived mode coupling coefficient is related to the XT and the amount of XT reduction (in dB) in TA-MCFs relative to normal step-index MCFs is formulated. In section 4, the results and discussions are presented. Firstly, XT estimations based on analytical expression and numerical simulations are compared, and it has been confirmed that the results obtained by our simplified analytical expression are in good agreement with those based on finite element method. Secondly, further reduction of XT in trench-assisted structure is described. Section 5 concludes the work.

Mode coupling coefficient formulations
The refractive index profiles for normal step-index and trench-assisted structures are shown in Figs. 1(a) and 1(b), respectively. The refractive indices for the core, cladding and trench are n 1 , n 0 and n 2 , respectively, where the refractive indices in the 1 st and 2 nd claddings in trench-assisted structure are assumed to be the same. The relative refractive index differences between core and cladding, trench and cladding are ∆ 1 and ∆ 2 , respectively. The core radius, the distance from the outer edge of the 1 st cladding to the core center, the distance from the outer edge of trench to the core center and trench width are a 1 , a 2 , a 3 and w tr , respectively. To have easy comparison, the core radius a 1 , the relative refractive index difference between core and cladding ∆ 1 in both structures are assumed to be the same.

Normal step-index MCFs
The mode coupling coefficient of directional coupler [13] is written as where ω is an angular frequency of the electromagnetic fields, ε 0 is the permittivity of vacuum, N 2 (x, y) denotes the refractive index distribution in the entire coupled region, N 2 q represents the refractive index distribution of waveguide q (which includes core q and cladding in MCFs), * denotes the complex conjugate and u z is a unit vector.
The mode coupling coefficient between two neighboring cores with step-index profiles in a directional coupler can then be simplified to [13] where U 2 1 = a 2 1 (k 2 n 2 1 − β 2 ) and W 2 1 = a 2 1 (β 2 − k 2 n 2 0 ), in which β is the propagation constant. V 1 = k a 1 n 1 (2∆ 1 ) 1/2 is the V number which determines the modes propagating in a fiber, where k = 2π/λ is the wave number and λ is the wavelength of light in vacuum. K 1 (W 1 ) is the modified Bessel function of the 2 nd kind with 1 st order and Λ is the core pitch (or core-to-core distance).
In order to compare with the mode coupling coefficient in a trench-assisted structure, formulated in Eq. (33) in [14], the mode coupling coefficient in normal step-index structure should also be expressed in a similar form. With Eq. (4.118a) in [13] and z = r/a 1 , the integral part of Eq. (4.117) in [13] can be written as where J 0 (U 1 r/a 1 ) is the Bessel function of the 1 st kind with zero order and I 0 (W 1 r/a 1 ) is the modified Bessel function of the 1 st kind with zero order, in which r is the radial distance from the center of the fiber core. Finally, the mode coupling coefficient of directional coupler with normal step-index profiles can also be expressed as where J 1 (U 1 ) is the Bessel function of the 1 st kind with 1 st order and K 0 (W 1 ) is the modified Bessel function of the 2 nd kind with zero order.

Trench-assisted MCFs
As the sizes of the 1 st cladding and the trench in trench-assisted structure are not infinitely large, the solutions in these two regions also contain the modified Bessel functions of the 1 st kind I 0 (W 1 r/a 1 ), which makes the derivation of the mode coupling coefficient in TA-MCFs difficult. If I 0 (W 1 r/a 1 ) in the 1 st cladding and the trench are ignored, the mode coupling coefficient between two adjacent cores in heterogeneous TA-MCFs can be written as Eq. (33) in [14], which can then be further simplified to the following for homogeneous case where L q , P 1 , P 2 , Y 1 , Y 2 and P 1 − P 2 + Y 1 − Y 2 can be expressed as For large arguments of ax and (a + b)x, the modified Bessel function of the 1 st kind can be approximated as If b is relatively small compared to a, I 0 (a + b)x can be related to I 0 (ax) by the following expression with good accuracy using the above relationship It should be noted that the approximation in Eq. (8) is valid only for large ax and (a + b)x, while Eq. (9) is also valid even when (a + b)x is small. Typical value of Λ (i.e. 30 µm) can be more than 5 times larger than trench width (i.e. 4.5 µm), thus ( ] > 0 is much smaller than W 1 /a 1 . Following Eq. (9), we have where Γ can be simplified to in which r has been dropped since ( is much smaller than W 1 /a 1 and r is much smaller than Λ. Substituting Eq. (10) into Eq. (7), the ratio between the mode coupling coefficients for the two structures becomes Assuming n = 1 in the dispersion equation Eq. (3.71) for step-index fiber in [13], thus Substituting Eq. (13) into Eq. (12), we get For x ≥ 2, the modified Bessel function of the 2 nd kind can be approximated as Thus, L q in Eq. (6) can be simplified to Finally, the ratio between the mode coupling coefficients for the two structures becomes Replacing κ pq with Eq.
(2), the mode coupling coefficient between two neighboring cores with trench-assisted structures in a directional coupler can then be expressed as 3. Relating mode coupling coefficients to inter-core crosstalk The statistical mean of the XT (mean XT, XT µ ) between two adjacent cores of a homogeneous MCF [15] can be expressed as where R b is the bending radius and L is the fiber length. Substituting Eqs.
(2) and (18) into the above expression, the analytical expressions for XT in normal step-index MCFs and TA-MCFs can be formulated, respectively.
For the estimations of XT reduction, the only difference between trench-assisted and normal step-index structures comes from the mode coupling coefficient, and the ratio of XT for the two different structures becomes where XT µ and XT µ are the mean XT in TA-MCFs and normal step-index MCFs, respectively. If the mean XT is expressed in dB, we have where 17.4 comes from 40 log 10 e. From Eqs. (9) ∼ (13) in [14], we have Since n 1 ≈ n 0 and it is assumed that |∆ 2 |/∆ 1 = m, we have V 2 2 ≈ mV 2 1 . For 1.5 ≤ V 1 ≤ 2.5 in step-index fibers, an excellent approximation of W 1 can be derived by assuming W 1 is a linear function of V 1 . It can be written as W 1 = 1.1428V 1 − 0.996, which is within 0.2% of accuracy over the range of V 1 [16]. Thus For 1.5 ≤ V 1 ≤ 2.5, W 2 can also be approximated to a linear function of V 1 , as listed in Table 1, where m = 0 means that there is no trench, which corresponds to normal step-index structure. The approximation accuracies at m = 1 and m = 2 are within ±0.3%.

Comparison with numerical simulations
In the previous derivation of the mode coupling coefficient in TA-MCFs, the modified Bessel functions of the 1 st kind I 0 (W 1 r/a 1 ) in the 1 st cladding and the trench have been ignored. Therefore, the XT obtained using Eq. (5) and the most rigorous finite element method (FEM) could be different. Furthermore, since the simplified analytical expression Eq. (18) is an approximation of the non integrable expression Eq. (5), comparison of the XT based on these two expressions is also needed. The comparison is done based on the structural parameters listed in Table 2, in which the distance from the outer edge of the 1 st cladding to the core center a 2 has been assumed to be twice the core radius a 1 . The effective area of each core will be reduced if the trench is placed too close to the core, while the trench should be placed as close as possible to the core to reduce core pitch for high core density MCFs. Taking these considerations into account, the ratio of a 2 /a 1 = 2 seems to be a reasonable choice.  The trench width w tr is now only limited by the core pitch and the distance between adjacent trenches since the trench should not overlap with the neighboring trenches, as shown in Fig. 2. The maximum cladding diameter (D clad ) is limited by mechanical reliability [17] since fibers with larger cladding diameters will experience larger tension due to bending. To satisfy the limit of failure probability, the cladding diameter should be less than 225 µm [9]. The cladding thickness (CT ) is the distance between the cladding edge and the center of the outermost cores, which is assumed to be at least 30 µm due to micro-bending loss consideration [8]. With these constraints and if the minimum distance between adjacent trenches is assumed to be 3 µm, the core pitch of the one-ring arranged 12-core MCF as shown in Fig. 2(a), should be less than 39 µm. Thus, the maximum distance from the outer edge of trench to the core center would be a 3 = (39 − 3)/2 = 18 µm. With the other parameters listed in Table 2, the allowed trench width w tr has to be less than 2a 1 (or 9 µm), where w tr = 4.5 µm is used in the table. The comparison of XT between analytical calculation and numerical simulations (both for non integrable expression and FEM) for TA-MCFs is plotted in Fig. 3. It should be noted that the XT presented here is only between two adjacent cores for simplicity without assuming any specific core arrangement in MCFs. The length and bending radius of the fiber are assumed to 100 km and 140 mm, respectively. The XT for normal step-index structure is also included for comparison, which can be regarded as trench-assisted structure with ∆ 2 = 0%.
In order to guarantee that the simplified analytical expression has good accuracy for ∆ 2 over a wide range, the XT at ∆ 2 of both − 0.35% and − 0.70% are included, where the relative refractive index difference between trench and cladding of ∆ 2 = − 0.70% might be the limit for regular VAD and OVD processes. The trench width should be large enough to reduce the XT and if the minimum trench width is set to be the same as a 1 as in Table 2, the corresponding core pitch would be 2a 3 + 3 = 2 × 3a 1 + 3 = 30 µm, in which 3 µm is the minimum distance between adjacent trenches. The numerical simulations based on the non integrable expression Eq. (5) and on FEM are carried out at core pitches of 30, 35, 40 and 45 µm, while the XT calculation using analytical expression Eq. (18) can be plotted over the entire continuous range.
As can be seen from Fig. 3, the XT estimations based on the analytical expression and on the non integrable expression are in good agreement as they almost overlap with each other. In other words, the approximation used to simplify the non integrable expression has very good accuracy. The XT simulated by FEM is always a little lower than the one obtained by the non integrable expression and the analytical expression. For the case at ∆ 2 of − 0.35%, the maximum XT difference between analytical expression and FEM is less than 0.7 dB, while for the case when ∆ 2 is changed to − 0.70%, this difference is increased a little, but is still less than 1.5 dB.
Thus the analytical expression can be used to estimate XT with good accuracy without the burden of numerical simulations based on FEM. As can be seen from Fig. 3, the XT curves versus core pitch at different ∆ 2 based on the analytical expression seem to be parallel to each other. Referring to Eq. (24), it is also found that the amount of XT reduction in trench-assisted structure relative to normal step-index structure is core pitch independent. As discussed previously, the maximum trench width can be larger than core radius depending on the allowed core pitch in high core density MCFs. The XT comparison is then extended to the case when w tr is larger than core radius a 1 , as shown in Fig. 4, in which the relative refractive index difference ∆ 2 between trench and cladding is set to − 0.70%. The numerical simulations based on the non integrable expression and FEM were performed at R tr = w tr /a 1 =1.0, 1.5, 2.0 and 2.5, while the XT estimation based on the analytical expression is plotted as a function of the ratio of trench width to core radius, where the XT at R tr = 0 corresponds to the normal stepindex structure. The core pitch in this comparison is set to be 45 µm for all different ratios of trench width to core radius, thus the distance between adjacent trenches is different for different values of R tr . With the maximum R tr of 2.5, the distance between adjacent trenches is calculated to be 4.5 µm (= 45 − 2 × (2 + 2.5) a 1 ), which still satisfies the minimum distance requirement between adjacent trenches of 3 µm.
Again, the XT estimations based on the analytical expression agree very well with those obtained by the non integrable expression and FEM at different ratio R tr of trench width to core radius. The XT based on FEM is still a little lower than the one calculated by analytical methods, but the maximum difference is below 1.5 dB for R tr ranges from 1.0 to 2.5, as plotted in Fig. 4. The maximum R tr used in the comparison is 2.5, which is realistic in practical applications. As the trench width is mostly limited by core pitch, and the core pitch in high core density MCFs is normally below 45 µm. It is important to note that the cable cutoff wavelength becomes longer when the trench width increases. For instance, with ∆ 2 of − 0.70%, the maximum trench width based on numerical simulations is 1.56a 1 so that the single mode operation is still guaranteed in C band (with corresponding cable cutoff wavelength of 1530 nm). By comparing with the FEM results, it is confirmed that the simplified analytical expression is valid for parameters such as ∆ 2 ranging from 0 to − 0.70% and trench width from 0 to as large as 2.5a 1 . It should be noted that the distance from the outer edge of the 1 st cladding to the core center is always set to twice the core radius in the above XT comparison.
Another issue for MCF transmission is the wavelength-dependent XT, which increases with wavelength. Since the transmission distance in MCFs is determined by the worst XT over the transmission band, it is important to understand the properties of wavelength-dependent XT for high capacity and long distance MCF transmission. TA-MCFs are widely adopted in capacity record transmissions [4, 5] because of their low XT, a theoretical study of the wavelengthdependent XT in TA-MCFs has already been carried out with our simplified analytical expression derived in this work, [12].

Prospect for further reduction of XT
The mean XT in trench-assisted structure, XT dB can be obtained by simply adding the term 10 log 10 Γ−17.4(W 2 −W 1 )w tr /a 1 to the XT in normal step-index structure XT dB as shown below XT dB = XT dB + 10 log 10 Γ − 17.4(W 2 − W 1 ) w tr a 1 .
Based on the above equation, it is easy to find that the XT in TA-MCFs can be reduced by increasing either W 2 − W 1 or R tr = w tr /a 1 . As derived previously, a larger ratio of relative refractive index difference m = |∆ 2 |/∆ 1 results in a larger W 2 − W 1 . As ∆ 1 in trench-assisted structure is set to be the same as in normal step-index structure for easy comparison, the XT reduction can be achieved by increasing the absolute value of the relative refractive index difference |∆ 2 | between trench and cladding. For instance, based on the parameters in Table 2, an increase of |∆ 2 | from 0 to ∆ 1 induces a XT reduction of 20.5 dB and an increase of |∆ 2 | from ∆ 1 to 2∆ 1 introduces a further XT reduction of 13.5 dB, as depicted in Fig. 3.
The relative refractive index difference between trench and cladding ∆ 2 is limited to be around − 0.70% by regular VAD and OVD processes. Other techniques of further reducing ∆ 2 have been proposed, such as hole-assisted structure [6]. As shown in Fig. 5, if ∆ 2 is reduced from − 0.70% to − 1.40%, a further XT reduction of around 21 dB is possible. It should be noted that an increase of |∆ 2 | or R tr may result in a longer cable cutoff wavelength, since the mode will be strongly confined in TA-MCFs. For instance, in order to have single mode operation in C band, the maximum trench width is only 0.96a 1 at ∆ 2 of − 1.40%, based on numerical simulations. If the ratio of refractive index difference m and the ratio of trench width to core radius R tr are fixed, XT in TA-MCFs can be reduced by increasing V 1 , as can be seen from Table 1. Since V 1 is the V number at the transmission wavelength, in order to have a higher V 1 , the theoretical cutoff wavelength (at which, V 1 = 2.405) should be shifted from the shorter wavelength towards the lower end of C band (1530 nm). In real applications, cable cutoff wavelength is more practically used, and it might be longer than theoretical cutoff wavelength, depending on trench volume (|∆ 2 | and w tr ) and core pitch. If cable cutoff wavelength (which must be shorter than 1530 nm to guarantee single mode operation if both C + L bands are used for high capacity transmission) is longer than theoretical cutoff wavelength, then V 1 will be smaller, which means the mode is less confined, resulting in worse XT.

Conclusion
An analytical expression for the mode coupling coefficient in homogeneous TA-MCFs is derived, which has a very simple relationship with the one in homogeneous normal step-index MCFs. With this analytical expression, the XT in MCFs with a trench-assisted structure can easily be calculated and plotted directly without the need of numerical simulations. Comparison with the most exact FEM simulations confirms that this analytical expression has very good accuracy, thus is very useful as a powerful design tool. This analytical expression can be used to quickly estimate the XT in trench-assisted MCFs and analyze the relationships between XT and various fiber parameters. Based on these analytical relationships and if the core pitch is fixed, it is found that the XT between adjacent cores in TA-MCFs is ultimately limited by the relative refractive index difference between trench and cladding, and the trench width due to core pitch limitation for high core density. The XT properties, such as XT dependence on core pitch and wavelength-dependent XT can be analyzed based on the analytical expression.