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Abstract
We propose a homogeneous five-mode twelve-core fiber with a trench-assisted structure, combining a low refractive index circle and a high refractive index ring (LCHR). The 12-core fiber utilizes the triangular lattice arrangement. The properties of the proposed fiber are simulated by the finite element method. The numerical result shows that the worst inter-core crosstalk (ICXT) can achieve at -40.14 dB/100 km, which is lower than the target value (-30 dB/100 km). Since adding the LCHR structure, the effective refractive index difference between LP21 and LP02 mode is 2.8 × 10−3, which illustrates that the LP21 and LP02 modes can be separated. In contrast to without the LCHR, the dispersion of LP01 mode has an apparent dropping, which is 0.16 ps/(nm·km) at 1550 nm. Moreover, the relative core multiplicity factor can reach 62.17, which indicates a large core density. The proposed fiber can be applied to the space division multiplexing system to enhance the fiber transmission channels and capacity.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the rapid development of the 5 G networks, artificial intelligence and cloud computing, the demand for data transmission continues to grow rapidly. In order to meet the increasing demand for data transmission, optical fiber communication technology is improving and developing constantly. The conventional single-mode fiber now has achieved a Shannon transmission limit of 100 Tb/s, which is already close to the physical limit of transmission due to nonlinear noise, fiber fusion damage phenomenon and amplifier bandwidth [1,2]. How to further improve the transmission capacity has become the main problem of the current research. Multiplexing technology has become a new research trend to increase the information transmission capacity, especially the space division multiplex (SDM). Based on the SDM technology, multi-core fiber (MCF), few-mode fiber (FMF) and multi-core few-mode fiber (MC-FMF) have been developed [3].
MC-FMF can transmit multiple signals simultaneously, which enables the capacity of optical fiber communication systems to be rapidly expanded by 2-3 orders of magnitude under the existing technology framework and solves the foreseeable capacity contraction problem in the future [4,5]. However, there will be huge inter-core crosstalk (ICXT) by mode or power coupling in the fiber, which must be reduced in the long-distance transmission process of MC-FMF. The reduction of ICXT can be achieved by increasing the core pitch. Nevertheless, increasing the core pitch will lead to a larger cladding diameter, which will reduce the mechanical stability of optical fiber [6]. It is noted that the cladding diameter is better to be lower than 225 µm [7]. In addition, the hole-assisted structure (HAS), the trench-assisted structure (TAS) and the heterogeneous structure (HS) can also suppress the ICXT [8–10]. The HAS has a particular problem that needs to be solved in actual fabrication: the air hole easily collapses. But this problem will not occur in the trench-assisted structure. Therefore, we adopt the trench-assisted structure to lower the ICXT.
This paper introduces a homogeneous 12-core fiber, which can support five linear polarization (LP) modes per core. And the proposed fiber has the characteristics of low ICXT, low differential mode group delay (DMGD) and bending resistance. COMSOL Multiphysics software based on the finite element method (FEM) is used to simulate the various fiber properties of this LCHR FM-MCF. The refractive index (RI) profile of LCHR and the triangular lattice arrangement of the fiber illustrate in Section 2. In Section 3, the coupled-power theory (CPT) is introduced, which is adopted to estimate the ICXT. Inter-mode crosstalk (IMXT) reflects by the effective refractive index difference (Δneff) between adjacent modes. In Section 4, the change of ICXT with Rb is shown. Bending Loss (BL) depending on the trench width and outer cladding thickness is discussed. Effective mode area (Aeff) variation with the fiber parameters can also be acquired. Furthermore, other properties including dispersion and DMGD as a function of wavelength are analyzed in Section 5. Finally, we summarize the properties of the optimal fiber. After simulation analyses, the proposed 12-core 5-LP modes fiber can be greatly applied to the SDM system to lower the IXCT and dispersion.
2. Design of the homogeneous LCHR 12-core 5-LP fiber
The cross-section of the proposed fiber and the RI distribution of the core are demonstrated in Fig. 1. We adopt the triangular lattice to arrange the cores. Cores 1 and 2 of the LCHR MC-FMF are marked as shown in Fig. 1(a). In the previously published papers [11–13] of MC-FMF, the typical RI distribution contains traditional step index and grade index. For the proposed structure, a low RI circle and a high RI ring are added to the core which can be seen in Fig. 1 (b). The high RI ring width is a, and the position of that main control the effective RI (neff) of LP21 mode. When the mode field diameter of LP21 is closed to high RI ring, the neff of LP21 increases. Conversely, the low RI circle r1 approaches the mode field diameter of LP02, the neff of LP02 reduces. Therefore, the low RI circle and high RI ring can increase the Δneff between LP02 and LP21 modes and reduce the IMXT, which can separate the two modes effectively during transmission. Meanwhile, the trench with a width of e is applied to suppress the ICXT. Thus, the proposed trench-assisted LCHR fiber can achieve low ICXT and ignored IMXT.
Fig. 1. (a) The cross-section view of the proposed 12-core fibers. (b) The profile of the LCHR core unit and refractive index distribution.
The initial parameters are depicted in Table 1. The core pitch (d) and the outer cladding thickness (OCT) determine the cladding radius. The cladding radius is calculated by:
For gaining low ICXT, the initial value of d is set to 45 µm. The OCT should not be too small, adding additional loss to the core [14]. Meanwhile, the dimension of the cladding also should be taken into account. Therefore, the cladding radius is chosen as 106 µm. The RI of the core, low RI circle, high RI ring and trench can be calculated in Table 1.
Table 1. The initial parameters of the proposed fiber
In addition to theoretical simulations, the actual manufacture of this structure should be considered seriously. Firstly, the fabricating methods of the optical fiber preform include plasma chemical vapor deposition (PCVD) and modified chemical vapor deposition (MCVD). The optical fiber made by MCVD has low loss, and the RI distribution of the optical fiber can be easily changed to make a variety of optical fibers [15]. However, the MCVD has the disadvantage of being complicated and having low deposition efficiency. The PCVD is a method that performs rod for oxidizing and depositing gas in a quartz tube by microwave plasma. Compared with the MCVD, the PCVD has higher deposition efficiency and lower deposition temperature [16]. Therefore, the PCVD is adopted to manufacture optical fiber preform for this proposed fiber. The filling material of different regions is shown in Fig. 1(b). The cladding and the inner cladding are formed by pure quartz glass (SiO2). The trench dopes with fluorine. And the core region including the high RI ring and low RI circle dope with different concentrations of GeO2. Considering the actual manufacture, the lowest RI difference between the trench and cladding (Δn2) is -0.7% [17]. To maintain low ICXT, the Δn2 is set as -0.7%. Next, the punch method is adopted to drill 12 holes in a pure silica rod. The doped rods are inserted into the pure silica rod. Finally, the 12-core fiber performs will be further melted and drawn. Therefore, we are sure that the proposed 12-core fiber can be manufactured.
3. Performance optimization of the proposed 12-core fiber
Since there are many parameters that affect the performance of optical fiber, the control variable method is used to calculate the variation law of all parameters, and the final parameters is determined according to the calculation results.
3.1 Inter-core crosstalk
The ICXT is a crucial factor that restricts the communication quality for MCF. The methods to calculate the ICXT are the coupled-mode theory and coupled-power theory [18]. Since the CPT has the characteristics of computation simplicity, it is adopted to compute the average ICXT of 12-core 5-LP mode fiber in this paper. For homogeneous MC-FMF, the ICXT with length L of fiber is written as [6]:
where $\bar{h}$mn represents the average power-coupling coefficient, it can be expressed as [6]: where κmn is the mode coupling coefficient and Rb represents the bending radius. The κmn is written as [19]:To ensure that the wished propagation modes can operate over the C + L band and transmit a long distance, the ICXT is better to be lower than -30 dB/100 km [20]. The proposed fiber can transmit five modes shown in Fig. 2, including LP01, LP11, LP21, LP02 and LP31 modes. To simplify the simulation process, the bending fiber is equivalent to the straight fiber, which can be expressed as [21]:
According to the above requirements, the dependence of ICXT on fiber parameters is analyzed by COMSOL software. The main factors affecting ICXT are core pitch, core radius, RI of core, size and RI of the assisted-structure. Fig. 3 exhibits that the change curve of ICXT varies with the fiber parameters. From Fig. 3(a), it is clear that the ICXT can reach the target value when a is larger than 1.4 µm. The ICXT is below -30 dB/100 km when r1 is greater than 2.9 µm in Fig. 3(b). Fig. 3(c) depicts the ICXT of LP02 mode and LP31 mode dropping sharply. The increased r can enhance the ability to limit light for the core, so the ICXT decreases. When r is more than 7.5 µm, the ICXT can meet the requirement. The dependence of ICXT on the trench e and the core pitch d is illustrated in Fig. 4. The wider the trench, the greater the ability to bind light energy. And from Eq. (3), we can know that ICXT is inversely proportional to d. Thus, as the e and d increase, the ICXT reduces. When e is greater than 6 µm and d is more than 44 µm, the ICXT of five modes can meet the target values, respectively. Fig. 5 demonstrates the relationship between the IXCT and RI. In Fig. 5, when Δn1, Δn3 and Δn4 are greater than 0.84%, 0.15% and -0.19%, the ICXT is less than -30 dB/100 km. The change of Δn1, Δn3 and Δn4 has little effect on ICXT.
Fig. 3. The variation of ICXT with (a) the width of high RI ring a, (b) the radius of low RI circle r1 and (c) the core radius r at 1550 nm (Rb = 80 mm).
Fig. 4. The relationship between ICXT and (a) the width of trench e and (b) core pitch d at 1550 nm (Rb = 80 mm).
3.2 Effective refractive index difference between adjacent modes
IMXT is usually reflected by Δneff between different modes in a core [22]. Several Refs. [23,24] have reported that when the value of Δneff is more than 1 × 10−3, the IMXT can be ignored. To make sure that the Δneff between adjacent modes meets the requirement, the high RI ring and the low RI circle are introduced. Fig. 6 shows the Δneff of adjacent modes varies with Δn1, Δn3 and Δn4. In Fig. 6(a), when the Δn1 is in the range of 0.8%-0.9%, the IMXT meets the requirement. And when the Δn3 is lower than 0.275%, the IMXT can reach the target value, as shown in Fig. 6(b). Moreover, the trend of the LP02-31 line is different from others, because the Δn3 rising is equivalent to an increase in RI for LP21 mode and LP31 mode. The LP02 mode is relatively unchanged. Thus, the Δneff between LP21 and LP02 mode rises, and that between LP02 and LP31 mode declines. Fig. 6(c) reveals that when the Δn4 is bigger than -0.175%, the Δneff is larger than 1 × 10−3. It can be seen in Fig. 6 that the Δneff between LP21 and LP02 and the Δneff between LP02 and LP31 change significantly when the low RI circle and high RI ring are added, indicating that the two structures have an impact on the neff of LP21 and LP02.
3.3 Bending Loss
As for the MCF, BL also can affect the performance of optical fiber, so it should be considered in the design optimization. To ensure 5-LP mode transmitting in the C + L band, the BL of LP31 mode (wished mode) should be lower than 0.5 dB/100turns at 1625 nm when the Rb is 30 mm. The BL of LP12 (unwished mode) should be more than 1 dB/m at 1530 nm when Rb is 140 mm [12]. The BL can be calculated by [25,26]:
It should be noted that the outer core is greatly affected by bending [15]. When the BL of the outer core satisfies the conditions, the BL of all cores can meet the requirements. Therefore, this paper selects the outer core to calculate the BL. The influence factors for BL are the trench structure and the OCT. The scan results are shown in Fig. 7. The BL reduces when the trench width increases. Fig. 7(a) shows the relationship between the width of the trench and BL. When e is less than 7 µm, the BL of LP12 at 1530 nm (Rb = 140 mm) is more than 1 dB/m. Meanwhile, when e is in the range of 5-9.5 µm, the BL of LP31 mode at 1625 nm (Rb = 30 mm) is far less than 0.5 dB/100turns. From Fig. 7(b), as OCT changes, the BL of LP31 mode at 1625 nm and LP12 mode at 1530 nm can satisfy the requirement. As a result, the LP31 mode can transmit and the LP12 mode will be cut off in this fiber when e is in the range of 5-7.5 µm and OCT is from 27 µm to 37 µm, respectively.
Fig. 7. The dependence of outer core bending loss of LP31 mode at 1625 nm (Rb = 140 mm) and LP12 at 1530 nm (Rb = 30 mm) on (a) the trench e and (b) the outer cladding thickness OCT.
3.4 Effective mode area
The Aeff is related to the nonlinear effect of the fiber. The signal distortion caused by the large nonlinear effect is severe, so the transmission fiber needs to have the Aeff as large as possible. In general, the effective area of the mode field is not less than 80 µm2 [13]. The Aeff can be calculated as [26]:
Fig. 8 shows the relationship between the core parameter and Aeff. Fig. 8(a) indicates that the Aeff change trend of LP02 mode is larger than other modes. The reason for that is the overlap area between the LP02 mode field and high RI varying with the position of the high RI ring. When the a is small, the overlap area is large. Then, as a increases the Aeff of LP02 mode reduces. Fig. 8(b) illustrates that the change of r1 mainly affects the LP02 mode. As the r1 increases, the low RI ring has a small overlap integral with the LP21 mode. While there is a large overlap integral with the LP02 mode. Thus, the Aeff of the LP02 mode significantly reduces, while the Aeff of the LP11, LP21 and LP31 mode is almost unaffected. Fig.s 8(b) and (c) reveal that when the r1 is below 3.7 µm and the a is more than 7.3 µm, the Aeff can be over 80 µm2. It can be found from Fig. 8(d) that the Aeff constantly becomes smaller as the Δn1 increases, but the change curve is relatively gentle and linear. The Aeff of the LP01 mode dramatically reduces, while the Aeff of the other modes is essentially unchanged in Fig. 8(f).
4. Summary of optimization results
Based on the above results, the optimized parameters are shown in Table 2.
Table 2. Optimized parameters of the LCHR 12-core 5-LP mode fiber
The ICXT of the five-LP mode between core 1 and core 2 at 1550 nm (Rb = 80 mm) is calculated, which is shown in Table 3. It can be found that the XT31-31 is the largest among the ICXT, which is -47.92 dB/100 km.
Table 3. The ICXT (dB/100 km) between core 1 and core 2 at 1550 nm (Rb = 80 mm)
Since there are several adjacent cores around each core, it is best to consider the sum of adjacent core crosstalk. And the adjacent cores around the inner core are more than the surrounding cores of the outer core, the inner core is may suffer a larger ICXT than the outer core. Thus, the ICXTworst of core 1 is calculated [7], which is shown in Table 4. It can be found that the ICXTworst is about 7.8 dB larger than the ICXT between core 1 and core 2, but the maximum ICXT is still lower than -30 dB/100 km. So, the ICXT of the entire fiber can meet the requirements of long-distance transmission,
Table 4. The ICXTworst (dB/100 km) of core 1 at 1550 nm (Rb = 80 mm)
In Fig. 9, as the wavelength varies, the ICXT and Δneff are also calculated. The ICXT of modes is less than -30 dB/100 km and the Δneff of adjacent modes are larger than 1 × 10−3 in the C + L band. The usual issue in reported articles is that the LP21 mode and LP02 mode are difficult to separate, which is solved in the proposed paper successfully. After adding a low refractive index circle and a high refractive index ring (LCHR) in the 12-core 5-LP mode fiber, the Δneff between LP21 and LP02 mode reaches 2.8 × 10−3 proves that LP21 and LP02 are separate. In conclusion, the low IXCT and ignored IMXT can ensure the MC-FMF maintains large-capacity long-distance transmission stably.
Fig. 9. (a) The relationship between ICXT and wavelength. (b) The relationship between Δneff and wavelength.
The bending of MCF will cause different changes in the RI of each core, which leads to a phase mismatch of adjacent cores. Phase mismatch will help to suppress crosstalk in homogeneous MCF. The bending radius of 500 mm can be approximately considered as the condition of straight optical fiber [6]. Fig. 10 shows the relationship between ICXT and Rb at 1550 nm. The Rb varies from 50 mm to 500 mm, the all ICXT values are lower than -30 dB/100 km. And when Rb is larger than 200 mm, the change amplitude of ICXT is not large. Therefore, the proposed fiber can be considered insensitivity to bend when Rb ≥ 200 mm.
5. Other properties of the LCHR 12-core fiber
5.1 Dispersion
The dispersion is a key factor in the long-distance and large-capacity optical fiber communication system. Low dispersion stands for lower signal distortion and less bit error rate [27]. And low dispersion slope signifies a larger band of flat dispersion region, which increases the available bandwidth of the channel. The dispersion coefficient can be calculated by [26]:
The LP01 dispersion of two types of fiber cores with different RI profiles as a function of wavelength is illustrated in Fig. 11. The red line represents the dispersion of the ordinary step-index core, while the green line is the dispersion curve of the proposed core. It can be seen that the dispersion of LP01 mode in the proposed model has a great drop, which demonstrates that the dispersion value can decrease effectively after adding the LCHR structure. Furthermore, the dispersion of other high-order modes is shown in Table 5.
Table 5. Optimal fiber properties of the LCHR 12-core 5-LP mode fiber at 1550 nm (Rb = 80 mm)
5.2 DMGD
Each mode in the few-mode fiber has a slight difference in transmission velocity, and their phase constants are different. The time delay will occur after transmitting the same distance at a specific frequency, which will gradually worsen with the increased distance. This delay is called the DMGD [12]. The DMGD between high-order modes (LPmn) and LP01 mode can be calculated as [15]:
Different DMGD modes will affect optical fiber performance in the current optical fiber transmission system. Large DMGD can keep MC-FMF with low ICXT and relatively independent transmission of each mode channel [6]. The relationship between DMGD and wavelength is shown in Fig. 12. It is apparent that with the increase of wavelength, the DMGD enlarges linearly in the C + L band. The DMGD of the high-order mode and the fundamental mode is less than 23 ps/m, which can contribute to reducing the difficulty of signal processing in the MIMO transmission system of FM-MCF [22]. The specific results of DMGD between high order mode and LP01 mode at 1550 nm (Rb = 80 mm) are shown in Table 5.
5.3 Relative core multiplicity factor
The core multiplicity factor (CMF) is defined as the size of the optical fiber mode field per unit area. It is an important parameter to measure the spatial multiplexing degree of multi-core fiber. The relative core multiplicity factor (RCMF) is employed to calculate the spatial efficiency of this fiber. The RCMF is the ratio of CMF between MCF and standard SMF (CD and Aeff are 125 µm and 80 µm2 at 1550 nm, respectively), which can be expressed as [6]:
The RCMF for this proposed LCHR trench-assisted twelve-core five-LP mode fiber can reach 62.17, which is larger than that 12-core reported in [28] and realizes high-density multiplexing. When Rb is 80 mm, the optimal fiber properties of the LCHR 12-core 5-LP mode fiber at 1550 nm are concluded, which is presented in Table 5.
6. Conclusion
In this paper, we propose a trench-assisted homogeneous LCHR 12-core 5-LP mode fiber with a triangular lattice arrangement. According to the numerical analysis results, the ICXT of modes satisfies the requirement in the C + L band. The ICXT is lower than -47 dB/100 km at 1550 nm (Rb = 80 mm). And the ICXT has little fluctuation when Rb is larger than 200 mm at 1550 nm, thus the proposed fiber is resistant to bending (Rb ≥ 200 mm). The Δneff between LP21 and LP02 mode reaches 2.8 × 10−3, which indicates that the separation problem of LP21 and LP02 is solved. And the Δneff of adjacent modes is larger than 1 × 10−3, so the IMXT can be ignored. The BL of LP31 mode is 1.19 × 10−7 dB/100 turns at 1625 nm and the BL of LP12 is 1.11 dB/m at 1530 nm, which testifies that LP01, LP11, LP21, LP02 and LP31 mode can operate over C + L band. And the Aeff of modes is more than 80 µm2 which can diminish the fiber nonlinear effect. Besides, compared with the ordinary step-index MCF, the dispersion of LP01 in the proposed RI distribution descends effectively. Based on various superior performances, the proposed fiber can be used for SDM systems to increase fiber channels and enhance transmission capacity.
Funding
National Key Research and Development Program of China (2019YFB2204001); National Natural Science Foundation of China (12074331).
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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