A Weakly-Coupled Few-Mode Optical Fiber With a Graded Concave High-Index-Ring

This paper proposes a novel refractive index profile design based on few-mode fibers (FMFs) which can support 4 linear polarization (LP) modes. We first present a FMF whose core is dually assisted by a nano-hole (NH) and a high-index-ring (HIR), and then substitute the dual assistance by an innovative graded concave HIR (GC-HIR) assisted structure. Using the finite element method (FEM), the parameters of the NH-HIR and GC-HIR optical fiber are adjusted to investigate their respective minimum effective refractive index difference (minΔneff), which is 2.012 × 10−3 for the former, and 2.532 × 10−3 for the latter. Both optical fibers have a significant improvement on the crosstalk suppression effect, and the GC-HIR optical fiber is even better. The proposed GC-HIR FMF with special refractive index profile has potential application prospects in mode division multiplexing (MDM) optical fiber communication system, and can provide theoretical foundation for the design and analysis of subsequent optical fibers and key components.


Introduction
With the development of optical fiber communication, multiplexing technology is widely used to improve communication capacity. However, due to the inherent nonlinear effect of single mode fiber (SMF), the system capacity is approaching the Shannon limit, thus traditional SMF no longer meet the development demand [1]. In order to solve the problem of insufficient system capacity, few-mode fiber (FMF) has become a research hotspot [2]- [5], and is considered one of the most straightforward optical fiber implementations for the short-reach space division multiplexing transmission system [6]- [9]. In general, FMFs have a simple structure that is easy to integrate [10]- [11], and need not to worry about the inter-core crosstalk as in multi-core fibers. However, inter-mode crosstalk in FMFs is a majority factor that can affect the communication quality.
As a solution to the inter-mode crosstalk, multiple input multiple output digital signal processing (MIMO-DSP) is a commonly adopted method, but with the increase of the mode number, the complexity and power consumption of the transmission system would increase greatly [12]- [14]. Another solution is to increase the effective refractive index difference ( n eff ) between adjacent modes through optical fiber design, which promotes the development of a variety of new optical fiber structures. n eff in cores is mainly related to the core refractive index (RI) profile, and relevant research shows that n eff ≥10 −3 [15] can effectively reduce the inter-mode coupling. For instance, Y. Xie et al. proposed an optical fiber featuring a nanopore-assisted step-index core with double-cladding structure, achieving a min n eff of 1.8 × 10 −3 [16]; S. Jiang et al. designed and fabricated a ring-assisted four-mode optical fiber with min n eff of 1.8 × 10 −3 [17]. R. M. Alexander et al. proposed an optical fiber of a depressed core and four high-index side holes, which increased the min n eff of 4 modes from 0.8 × 10 −3 to 1.2 × 10 −3 without significantly affecting other characteristics [18].
In this paper, we first present a FMF which can support 4 linear polarization (LP) modes and whose core is dually assisted by a nano-hole (NH) and a high-index-ring (HIR), and then substitute the dual assistance by an innovative graded concave HIR (GC-HIR) assisted structure. Using the finite element method (FEM), the parameters of the NH-HIR and GC-HIR optical fiber are adjusted to investigate their respective min n eff . Calculation shows that both optical fibers have a significant improvement on the min n eff , and the latter has an even larger value, thus achieving a better crosstalk suppression effect. The two proposed FMFs, especially the GC-HIR optical fiber, has the characteristics of supporting multiple modes and simple preparation, which can effectively solve the problem of inter-mode crosstalk in mode division multiplexing (MDM) systems, demonstrating an important application value in high-speed and large-capacity optical fiber communication system.

Structural Design of Proposed Optical Fibers
According to the coupled mode theory, a large n eff between adjacent modes help suppress intermode crosstalk [19], [20]. Considering the current implementation of the reported works, we first come up with a dually assisted NH-HIR optical fiber by combining together the nano-hole and the high-index-ring. The nano-hole assistance is to penetrate a nano-scale air-hole through the core of the step-index optical fiber, whose low RI can fully reduce the n eff of LP 02 without significantly influencing other modes. The ring assistance is to add a high-index-ring at a position where the power of the LP 02 mode is the lowest, thus its overlapping integral with LP 02 is small while that with LP 21 mode large. The parameters of this NH-HIR optical fiber will be adjusted to investigate its min n eff to confirm the effect of the dual assistance.
Based on the parameter adjustment result in the first step, the NH-HIR dual assistance will be further substituted by a graded concave HIR assisted structure. A step-index FMF with the same core radius and relative RI difference n will be used as the basis for GC-HIR design. Within the fiber core of the step-index FMF, a special ring of circularly symmetrical graded RI distribution is introduced. For this ring, its RI value increases gradually along the ring radius, forming a radiation RI distribution. The ring edge has the highest RI, whose function is similar to that of the high-index-ring in the NH-HIR optical fiber structure; the ring center has the lowest RI (still higher than that of the cladding), whose function is similar to that of the nano-hole. By resorting to a graded RI distribution assistance within the fiber core, similar crosstalk suppression effect is achieved, or even better. Fig. 1 illustrates the transverse cross-section and RI distribution of the conceived NH-HIR optical fiber (a) and GC-HIR optical fiber (b). For the NH-HIR optical fiber, the cladding was made of pure silica with RI n 2 = 1.444; the core with a radius a = 7.5 μm was made of Ge-doped silica, whose RI n 1 = 1.4537; the RI of high-index-ring is n_ring; considering the preparation process, the fiber cladding diameter D was designed to be 125 μm, consistent with a standard SMF; the relative RI difference of the unassisted part is n = (n 1 2 -n 2 2 ) / 2n 2 2 = 0.67% and that of the HIR is n + = (n_ring 2 -n 2 2 ) / 2n 2 2 -n = 0.2%; the radius of the nano-hole was indicated by R_hole, the inner diameter and width of the HIR by R_ring and W_ring, respectively. The HIR of circularly symmetrical graded RI distribution in the GC-HIR optical fiber was made of Ge-doped silica of different doping concentration, and its initial radius b is determined by R_ring and W_ring of the NH-HIR structure, namely, b = R_ring + W_ring. The highest doping concentration at the ring edge is indicated by P 1 , and the lowest at the ring center by P 2 . The RI of the Ge-doped silica can be given by the following composite Sellmeier equation [21]: where SA, Sl, GA, Gl are the coefficients of Sellmeier equation of SiO 2 and GeO 2 , respectively, and P is the mole percentage of GeO 2 , here representing the doping concentration of core part without assistance.

Parameter Optimization
In order to investigate the min neff, the parameters of NH-HIR optical fiber are optimized using FEM and further applied to designing the GC-HIR optical fiber. Fig. 2 shows the respective influence of R_ring, W_ring and R_hole on the modal n eff (the left column) and n eff (the right column) at the wavelength of 1550 nm with n + = 0.2%. First, the evolution of n eff with the change of R_hole from 0.1 μm to 0.5 μm is presented by Fig. 2(a). We can see that the introduction of nano-hole hardly affects the n eff of LP 01 and LP 02 . However, compared with the former, the latter has a relatively more obvious curvature. The corresponding change of n eff with the increase of R_hole is shown by Fig. 2(b). It can be found that the R_hole size has a significant effect on the n eff between LP 21 and LP 02 and that between LP 01 and LP 11 , where the former exhibits a great increase while the latter an obvious drop. When R_hole = 0.15 μm, the two n eff curves intersect, indicating a well-balanced modal min n eff . Similarly, by respectively investigating the variation of n eff with the change of R_ring and W_ring, the following values were obtained: R_ring = 3 μm, and W_ring = 1.9 μm. In sum, when R_hole = 0.15 μm, R_ring = 3 μm and W_ring = 1.9 μm, the inter-mode n eff of the NH-HIR optical fiber has the minimal value. Consequently, the inital value of the ring radius b in the GC-HIR optical fiber is set to be b = R_ring + W_ring = 4.9 μm. Based on the adjusted parameters of the NH-HIR optical fiber, the initial parameters of the GC-HIR optical fiber are preliminarily set, and further adjusted to investigate the min n eff . The change curve of n eff with b varying around its initial value 4.9 μm (4.5 ∼ 5.5 μm) is shown in Fig. 3(a). We can see that the gap between the n eff of LP 21 and LP 02 gradually enlarges and the gap between the n eff of LP 01 and LP 11 gradually shrinks with the increase of b from 4.5 to 5.5 μm. The corresponding change of n eff with b is shown by Fig. 3(b). It is clear that the radius of GC-HIR part b has a significant effect on the n eff between LP 21 and LP 02 and between LP 01 and LP 11 . When b = 5 μm, a well-balanced inter-mode min n eff was achieved. The influence of the doping concentration P 1 and P 2 on n eff and n eff is respectively evaluated, as shown by Fig. 3(c) to Fig. 3(f). As a result, the following final values were obtained: P 1 = 0.132, and P 2 = 0.061. According to the aforementioned composite Sellmeier equation, the core refractive index is 1.4637 and 1.4532, and the corresponding core-cladding relative refractive index difference is 1.364% and 0.637%, respectively. In sum, when b = 5 μm, P 1 = 0.132 and P 2 = 0.061, the inter-mode n eff of the GC-HIR optical fiber has the minimal value.

Advantage in Inter-mode Crosstalk Suppression
In order to highlight the advantages of our proposed GC-HIR structure, the inter-mode n eff of step-index optical fiber, NH assisted optical fiber, HIR assisted optical fiber, GC-HIR optical fiber, and the NH-HIR optical fiber is respectively investigated, as presented by Fig. 4. For all these optical fibers of different structure with the same core radius (a = 7.5 μm), except the min n eff , their RI distribution profiles and corresponding cross-section diagrams are demonstrated as well in Table 1.
From the comparison, we can see that the min n eff of ordinary step-index optical fiber is 0.8 × 10 −3 , which is only 10 −4 orders of magnitude. The min n eff between adjacent modes with only NH assisted or HIR assisted structure is about 1.4 × 10 −3 . The proposed NH-HIR dually assisted structure increases its min n eff to 2.012 × 10 −3 , 2.5 times than that of the original step-index optical fiber, which greatly helps inhibit the inter-mode crosstalk. Finally, the innovatively proposed GC-HIR optical fiber achieves the largest inter-mode min n eff , that is, 2.532 × 10 −3 , proving its better performance in crosstalk suppression. Therefore, by introducing the GC-HIR assisted structure into the core design, the inter-mode crosstalk has been greatly reduced. Without increasing the MIMO complexity, just the fiber core design has already met the requirement of low modal crosstalk, and all supported modes can be used as different transmission channels more easily.

Other Performance Investigation of the GC-HIR Optical Fiber
The electric field distribution diagrams of each mode supported by the GC-HIR optical fiber is shown in Fig. 5. Due to the special RI distribution of the core, LP 01 and LP 02 are significantly influenced by the low RI part, while other modes are almost unaffected. We further investigate the wavelength dependence of some important performance indices at C+L band. Table 2 presents the structural parameters of the proposed GC-HIR optical fiber and related performance indices.
A eff is defined by a quantitative measurement of the transverse area occupied by a mode in the waveguide or optical fiber. The calculation results show that the smallest A eff is greater than 40 μm 2 , thus the high-order mode would have an even larger A eff , and would not have to worry about excessive nonlinear effects.
Calculation shows that the dispersion of the 4 LP modes within the entire C+L band is at the range of 0 to 20, which is equivalent to that of a standard SMF (17 ps/nm/km in the ITU-T).  Therefore, the GC-HIR optical fiber proposed here can be applied to short-distance large-capacity optical networks.
DMD directly affects the complexity of MIMO signal processing [22], [23], which is defined as the group delay difference between the higher-order mode LP mn and the fundamental mode LP 01 . For the GC-HIR optical fiber, the DMD increases slightly from short wavelength to long wavelength, and its broadband characteristics meet the requirements of working at the C+L band.

Comparison With FMFs Released Recently
The characteristics of the two optical fibers designed in this paper (the NH-HIR and the GC-HIR optical fibers) are compared with other similar FMFs recently released, as shown by Table 3. From the comparison, we can see that our two optical fiber designs can achieve a larger min n eff , and the crosstalk between adjacent modes are better suppressed. Both of these two designs, especially the novel the GC-HIR optical fiber, meet the transmission requirements of the high-speed and large-capacity MDM system, and provide theoretical foundation for the design and analysis of subsequent optical fibers and key components.

Conclusion
In this paper, a FMF with its core dually assisted by a nano-hole and a high-index-ring is designed, which works as the initial parameter supplier and contrast reference for the later proposed GC-HIR optical fiber. By replacing the dual assistance by the graded concave HIR structure, the min n eff between adjacent modes can reach 2.532 × 10 −3 , achieving an even better inter-mode crosstalk suppression. Furthermore, the mode field distribution of the GC-HIR optical fiber is simulated, and main performance indices, including n eff , n eff , A eff , DMD and loss of each mode, are also analyzed over the whole C+L bands. The proposed GC-HIR FMF can effectively solve the problem of intermode crosstalk in high-speed large-capacity MDM system, demonstrating important application value, and can provide theoretical foundation for the design and analysis of subsequent optical fibers and key components.