Weakly guiding graded-index ring-core fiber supporting 16-channel long distance mode division multiplexing systems based on OAM modes with low MIMO-DSP complexity

Xi Zhang; Xi Zhang; Shi Chen; Shi Chen; Jian Wang; Jian Wang

doi:10.1364/OE.464559

1. Introduction

Optical fiber communication supporting voice, video, and data transmission is the backbone of telecommunications infrastructure. The demand for larger data volumes with the arrival of the 5G era yields new challenges to the global networks. A critical task in optical communications research is to find a way to further increase the transmission capacity of optical fiber communication systems. Space-division multiplexing (SDM) technology utilizing core division multiplexing (CDM), MDM or combination of CDM and MDM in multi-core fibers (MCFs) and multi-mode fibers (MMFs) has been extensively studied, which is considered as a promising approach for increasing the per-fiber capacity [1–4]. For CDM technology, the number of core channels in a single fiber strand has an upper limit owing to the limitation of mechanical reliability and inter-core crosstalk, while MDM technology has infinite orthogonal mode channels theoretically, which has led the researchers to explore a series of novel MDM transmission technology based on different mode bases such as linearly polarized (LP) modes and OAM modes [5–10]. In particular, OAM relating to the spatial phase structure of photons, has shown its great possibility for applications in both free-space and fiber-based transmission links. An OAM beam features a helical phase front of exp(ilθ) and has the discrete value of lħ per photon [11,12], where l is the topological charge value, representing the twisting rate of the spiral phase, θ refers to the azimuthal angle, and ħ is the reduced Planck constant.

However, with the number of multiplexed modes and transmission length increasing in the fiber transmission system, the mode coupling and differential mode delay (DMD) become unignorable which make the MIMO-DSP indispensable at the receiver [13,14]. To the best of our knowledge, RCFs supporting OAM modes with single radial mode order possess natural advantage for high-capacity MDM transmission with low MIMO complexity [15,16]. Crosstalk between different MGs can be reduced by method of strengthening separation of refractive indices between MGs, and this kind of RCF with low inter-group crosstalk can be utilized in MIMO-free transmission systems [17,18]. Besides, in the case of intra-group mode multiplexing, the modes in the same MGs have relatively large crosstalk, which should be mitigated by small-scale MIMO-DSP technology, e.g. 2 × 2 or 4 × 4 MIMO-DSP [19]. On the other side, the relatively high-contrast-index of RCF with large inter-group crosstalk may bring in manufacturing difficulty and large fiber loss [20]. In this scenario, a laudable goal in OAM-MDM communications would be to design a weakly-guiding RCF with multiple mode channels, low loss and low MIMO-DSP complexity.

In this paper, we propose a weakly-guiding graded-index RCF with trench-assisted structure supporting 8 OAM MGs. The cladding of designed RCF adopts fluorine-doped silica which is beneficial to reduce the fiber loss. Besides, more mode channels are conducive to expanding single-core capacity. The low inter-group crosstalk (< −25 dB/km) and low intra-group DMD (< 125 ps/km) of higher order OAM MGs (topological charge |l| = 4, 5, 6, 7) verify that the designed RCF has great potential for supporting 16-channel long-distance MDM transmission with low MIMO-DSP complexity. Moreover, we also give the comprehensive characterization of OAM modes in the designed RCF over the whole C band and evaluate its tolerance to ellipticity and bending. The designed RCF shows good characteristics, and can be employed in small-scale MIMO-DSP assisted intra-group modes multiplexing transmission or MIMO-free inter-group modes multiplexing transmission compatible with existing wavelength division multiplexing (WDM) technique.

2. Concept

2.1 Chromatic dispersion (D_λ) and differential mode delay

In the characterization of OAM modes in optical fibers, the chromatic dispersion and DMD associated with the l-th eigenmode should be calculated, which are expressed as follows [21,22]:

(1)$${D_\lambda } = {{ - \lambda } / c} \cdot {d^2}{{{n_{eff}}} / {d{\lambda ^2}}}$$

(2)$${\tau _l} = {{ - {\lambda ^2}} / {2\pi c}} \cdot {{d{\beta _l}} / {d\lambda }}$$

(3)$$DM{D_{lm}} = {\tau _m} - {\tau _l}$$

where c is the light velocity in vacuum, λ is the wavelength in vacuum, n_eff is the effective refractive index (RI), β is the phase constant of the fiber eigenmode, respectively. Here, DMD, as a crucial parameter for RCF’s transmission, determines the memory length of the optical transmission system and subsequently the computational complexity of MIMO-DSP [23].

2.2 Crosstalk (XT)

When calculating the inter-group XT, micro-bending resulting from the high frequency random perturbations of the fiber core along the fiber axis is considered. According to the analysis in [24,25], micro-bending-induced power coupling coefficient of adjacent MGs with single radial order can be expressed as

(4)$$2{\gamma _{lm}} = \frac{{Ck_0^2{\sigma ^2}}}{{2[{1 + {{({\varDelta \beta {L_c}} )}^{2p}}} ]}}\frac{{{{\left[ {\int_0^b {r\frac{{\partial n}}{{\partial r}}{A_l}{A_m}dr} } \right]}^2}}}{{\int_0^b {{{|{{A_l}} |}^2}rdr\int_0^b {{{|{{A_m}} |}^2}rdr} } }}$$

where $C = {\left[ {\mathop \smallint \limits_{ - \infty }^\infty \frac{1}{{1 + {{({\Delta \beta {L_c}} )}^{2p}}}}d\Delta \beta } \right]^{ - 1}}$. k₀ is the wave number in the vacuum. $\Delta \beta = 2\pi \Delta {n_{eff}}/\lambda $ is the propagation constant difference between the coupling modes l and m. A_l and A_m are the normalized amplitudes of mode l and m, respectively. σ and L_c are the standard deviation and correlation length of the perturbations, respectively. A typical value of p is 1, 2, or 3 lying on the external perturbation and fiber fabrication processing. r and n stand for the fiber radius and the refractive index, respectively. We first use full-vector finite element method (FVFEM) to calculate the propagation constant difference Δβ and the normalized amplitudes A_l and A_m, combined with the standard deviation and correlation length of the perturbations which are preset according to the simulation results and manufacturing experience, the required crosstalk can be achieved. The first molecular formula on the right side of Eq. (4) represents perturbation in transmission direction, whose physical significance is the spectrum distribution of the interaction between mode and fiber perturbation in the transmission direction. In the same way, the second molecular formula relates to horizontal perturbation standing for the mode coupling caused by fiber perturbation in the cross section of the fiber. From Eq. (4), one can clearly deduce that there are three major approaches to decrease the power coupling coefficient. First of all, the inter-MG Δβ should be maximized by modulating refractive index profile (RIP). Secondly, the horizontal overlap integration between the modal fields and refractive index gradient should be reduced simultaneously. Thirdly, the fiber manufacturing process needs to be optimized to improve σ and L_c which are strongly influential to the crosstalk. In this paper, we fix σ = 0.03 µm, p = 3 and L_c = 1.4 mm for crosstalk simulation.

2.3 MIMO algorithmic complexities

Based on time domain equalization (TDE), the tap length of each finite impulse response (FIR) filter (equalizer length) is given by [26]:

(5)$${N_{tap}} = {R_s}\Delta \tau LB$$

where R_s is sampling rate (samples/symbol), Δτ is the DMD, L is the fiber length and B is the symbol rate (or baud rate).

The total computational complexity to recover all symbols at once by MIMO equalization algorithms is given by:

(6)$${C_{TDE}} = D_s^2({R_s} + 1)\Delta \tau LB$$

where D_s is the multiplexed channel number, thus, the mode transfer matrix H(n) is D_s× D_s matrix.

2.4 Refractive index profile (RIP)

The modal properties of the RCF strongly depend on the RIP. It is worth noting that an important feature of graded-index designs is the reduced DMD by up to 2 orders of magnitude compared to step-index designs [27]. Furthermore, a trench-assisted cladding structure can also reduce the DMD value and improve the fiber macro-bending performance [28]. Therefore, we design a graded-index RCF with a trench-assisted cladding structure to modulate RIP for meeting the demand of low DMD. Figures 1(a) and 1(b) show the schematic cross-section and RIP of the designed weakly-guiding graded-index RCF, which is composed of a germanium-doped-silica ring core, fluorine-doped-silica cladding and trench with different dopant concentrations. The inner and outer radii of the fiber ring core are denoted as r₁ and r₂, respectively. In the same way, r₃ and r₄ represent the inner and outer radii of the trench, respectively. The cladding radius is set at 62.5 µm. As we know, decreasing refractive index of ring core and adopting fluorine-doped-silica cladding are beneficial to achieve low loss in RCF [29]. Assumed that n₁ is the maximum RI of ring core, n₂ equaling 1.444 is the RI of pure SiO₂ at 1550 nm, n₃ is the RI of fluorine-doped-silica cladding, n₄ is the RI of trench, Δn₁ = (n₁- n₃)/n₃ is the RI difference between ring core and cladding, Δn₂ = (n₂- n₃)/n₃ is the RI difference between pure SiO₂ and cladding, Δn₄ = (n₂- n₄)/n₄ is the RI difference between pure SiO₂ and trench, and g is the graded-index RIP parameter determining the shape of the RI gradient curve. By modulating the above parameters, fiber design with target excellent performance can be obtained.

Fig. 1. (a) Schematic cross-section and (b) refractive index profile of the designed fiber.

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3. Fiber design

In order to realize the application of high-capacity long-haul optical transmission system, we intend to adopt MDM technology based on OAM modes utilizing intra-group modes assisted with small-scale MIMO-DSP and inter-group modes without MIMO-DSP. In this case, our designed weakly-guiding RCF should meet the following targets: 1) the width of ring core should be optimized to suppress radial high-order modes for simplified scale of four modes in each OAM MG and subsequently simplified MIMO-DSP, 2) the inter-MG Δn_eff should be large enough to suppress inter-group coupling and keep low crosstalk between MGs, 3) the dopant concentration of fiber should be relatively low which ensures the low fiber loss to support long-haul transmission, 4) the intra-MG DMD should be reduced to improve the memory length of the optical transmission system and subsequently reduce the computational complexity of MIMO-DSP at the receiver in optical transmission lines.

Considering the traditional fiber manufacture facility restriction, the molecular fraction doping of GeO₂ and F should be low to ensure low relative refractive index difference between core and cladding. Moreover, to simplify the process of fiber design, the maximum core-to-cladding relative RI difference (Δn₁), cladding-to-pure-silica relative RI difference (Δn₂) and trench-to-cladding relative RI difference (Δn₄) are set to 1%, 0.3% and 0.7%, respectively. The parameter g representing RI gradient of ring core is fixed at 2 or 4. To determine the value of other fiber parameters of the designed weakly-guiding graded-index RCF with trench-assisted structure shown in Fig. 1(b), we first use FVFEM to optimize r₁ and d₁ (d₁= r₂ - r₁) with fixed d₃ (d₃= r₃ - r₂) and d₄ (d₄= r₄ - r₃) both equaling 2 µm which are the compromised sizes of trench structures. The calculated mode number as function of r₁ and d₁ with different g are illustrated in Figs. 2(a) and 2(d). It is quite clear that the mode number shows trend of accelerated growth with r₁ and d₁ increasing. Furthermore, to support the identical number of modes, the larger is the inner core radius, the smaller is the width of ring core. Under the circumstance of mode number being 30, the intra-group DMD of low-order MGs (|l| = 0, 1, 2, 3) is pretty small but the inter-group crosstalk among adjacent MGs is too large (>-15 dB/km) to adopt MDM technology utilizing intra-group modes assisted with small-scale MIMO-DSP. For OAM MGs with |l| = 4, 5, 6, 7 which show great potential for either mode group multiplexing or small-scale mode multiplexing, the maximum intra-group DMD appears in the 7^th order MG and maximum inter-group crosstalk appears between 4^th and 5^th order MGs, which as functions of inner radius (r₁) and width (d₁) of ring core with g = 2 or 4 are shown in Figs. 2(b)–(c) and 2(e)–(f). The white region is where the fiber supports more or less than 8 MGs which is beyond our consideration. It can be seen from Figs. 2(b) and 2(c), fixing r₁, the wider the d₁, the larger the DMD and the smaller the XT, which are due to the unignorable difference between modal intensity distribution of different MGs under the large ring width. To meet the requirement of low inter-group XT and low intra-group DMD, we make a compromise choice that point g = 4, r₁= 11 µm and d₁= 3.5 µm in the black circle are the final design.

Fig. 2. (a) The calculated mode number, (b) intra-group DMD of 7th order MG, and (c) inter-group crosstalk between 4th order and 5th order MGs as functions of r₁ and d₁ with g = 2, d₃ = d₄ = 2 µm; (d) The calculated mode number, (e) intra-group DMD of 7th order MG, and (f) inter-group crosstalk between 4th order and 5th order MGs as functions of r₁ and d₁ with g = 4, d₃ = d₄ = 2 µm.

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Moreover, to determine the structure size of the trench outside the ring core, we sweep inner radius (r₃, from 14.8 µm to 17.4 µm with 0.2 µm spacing) and width (d₃, from 1 µm to 3 µm with 0.2 µm spacing) with g = 4, r₁= 11 µm and d₁= 3.5 µm to calculate the crosstalk between adjacent mode groups and intra-group DMD at 1550 nm by full-vector finite-element mode solver. The calculated intra-group DMD of 7^th order MG and inter-group XT between 4^th and 5^th order MGs as functions of r₃ and d₃ are shown in Figs. 3(a) and 3(b). To achieve relatively low intra-group DMD and inter-group crosstalk at the same time, we eventually select the point r₃= 15.8 µm and d₃= 2 µm in the green circle as the target trench size.

Fig. 3. (a) Intra-group DMD of 7th order MG, and (b) inter-group crosstalk between 4th order and 5th order MGs as functions of r₃ and d₃.

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4. Properties and discussions

4.1 Modal properties

We then provide mode properties for the target fiber structure design. The simulated intensity profiles and spatial phase distributions of guided OAM modes in the designed RCF are shown in Fig. 4. One can clearly see the doughnut intensity profiles due to the phase singularity at the beam center.

Fig. 4. The intensity and phase distributions of OAM modes supported in the designed fiber.

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Fig. 5. (a) Calculated mode number and min(Δn_eff) between adjacent high-order MGs (|l| = 4, 5, 6, 7). (b) Crosstalk between adjacent high-order MGs. (c) Chromatic dispersion, (d) intra-group DMD, (e) A_eff, (f) nonlinearity of different modes versus wavelength over the C band.

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4.2 Fiber application scenario

To give a specific application scenario (transmission distance, transmission loss, MIMO complexity, etc.) of the designed RCF without fabricating an optical fiber, it is necessary to carry out an application feasibility analysis in combination with the fiber manufacturing process and previous experimental work. There are some inevitable differences between the fabricated fiber and the designed fiber due to the errors introduced by fiber manufacturing process, which would influence the transmission loss and modal performance of the fiber. Excitingly, the existing fiber manufacturing process is becoming more and more accurate and stable, and applicable for our slightly complex designed RCF.

On the other hand, fiber transmission distance is closely related to transmission loss, mode effective RI difference (Δn_eff) and crosstalk. It is not difficult to understand that the lower the fiber transmission loss, the longer the signal transmission distance. Moreover, larger Δn_eff between MGs benefits smaller mutual influence and crosstalk among the MGs, and subsequently reduces degree of distortion of the modes from different MGs after transmitting a certain distance. In other words, the modes in different MGs with larger Δn_eff can transmit a longer distance without increasing the difficulty of signal demodulation. In the previous work [30], using two high-order OAM modes (OAM₃₁, OAM₄₁) each carrying a quadrature phase-shift keying signal, the authors demonstrated a record long-haul OAM multiplexing transmission over 300 km RCF assisted by a recirculating loop without MIMO equalization. The single span of the RCF is 50 km, and the cladding is fluorine-doped. For this kind of step-index RCF, we have carried out a more detailed test [29]. The step-index RCF supports a total of 22 modes (6 MGs), and the tested transmission loss of each mode is less than 0.25 dB/km, and the tested mode crosstalk between OAM₃₁ and OAM₄₁ is -23.7 dB/km. In addition, according to simulation results based on the tested RIP, the Δn_eff between OAM₃₁ and OAM₄₁ is larger than 1.07×10⁻³.

While in our target weakly guiding graded-index RCF, Table 1 lists the inter-group Δn_eff, inter-group XT and intra-group DMD of four OAM MGs with |l| = 4, 5, 6, 7 at the wavelength of 1550 nm. Note that the Δn_eff and XT between adjacent MGs are larger than 1.18×10⁻³ and lower than −25.3 dB/km, respectively, indicating fully lifted OAM MGs channels and better performance than the step-index RCF, which mean the single-span transmission distance for our designed RCF can be higher than 50 km. On the premise that the fiber manufacturing process can be maintained at a normal level, the length of a single span for optical transmission can even reach 100 km. Since the designed RCF is a weakly-guiding fiber with fluorine-doped cladding and low doping concentration which is beneficial to achieve low fiber transmission loss, same as the step-index RCF in our previous work, the expected transmission loss of this designed fiber will not be larger than 0.25 dB/km. Combined with a recirculating loop, record 500-km OAM multiplexing transmission utilizing this kind of RCF is expected to be realized.

Table 1. The inter-group effective RI difference (Δn_eff), inter-group XT and intra-group DMD of four OAM MGs with |l| = 4, 5, 6, 7 in the target weakly-guiding graded-index RCF at 1550 nm

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4.3 MIMO algorithmic complexities

Equations (5) and (6) show that different B, different L and R_s have different requirements for the equalizer length. In the same way, the total computational complexity of MIMO equalization algorithms has a definite value only when R_s, B, and L are determined. Nevertheless, based on these equations, we can give the advantage for transmission of our designed RCF compared with traditional fiber. As for the designed weakly-guiding graded-index RCF supporting 30 eigenmodes divided into 8 MGs, the Δn_eff between the 5th to 8th MGs (|l| = 4, 5, 6, 7) are all larger than 1.18×10⁻³, enabling MIMO-free inter-group modes multiplexing. Hence, the traditional 16×16 MIMO equalization is simplified to 4 sets of 4×4 MIMO equalization. Assuming that the intra-group DMD and inter-group DMD are Δτ_d and Δτ, respectively. The intra-group DMD is lower than 125 ps/km for each MG and inter-group DMD between MGs (|l| = 4, 5, 6, 7) is 1.419×10⁴ ps/km, which mean Δτ /Δτ_d ≈114. In the time domain, the modified MIMO equalization needs Δτ_dR_sLB equalization length and 16Δτ_d(R_s+1)LB complex multiplications in total according to Eqs. (5) and (6), while in comparison, the traditional 16×16 MIMO equalization needs ΔτR_sLB equalization length and 256Δτ(R_s+1)LB complex multiplications, which are around 114 and 1824 times larger than the modified MIMO equalization. Thus, by combination of intra-group and inter-group MDM, the designed RCF shows potential to support 16-channel long distance MDM systems with low MIMO-DSP complexity.

4.4 Tolerance to bending and ellipticity

Ideal fiber is difficult to fabricate due to the imperfections and external perturbations (e.g. stress/strain, extrusion, twist, bend, temperature) introduced by the actual manufacturing process. For instance, stress/strain and extrusion can cause ellipticity in the fiber, leading to the birefringence between degenerate eigenmodes of the fiber and subsequently polarization mode dispersion. Twisting and bending are also the important contributors to fiber birefringence. Those imperfections may exacerbate the difference of propagation constants between the two fiber eigenmodes forming the OAM modes, which affect the mode profile and purity of the synthetic OAM modes, and induce modal coupling and crosstalk between OAM modes. For this reason, evaluating the effects of ellipticity introduced by stress and fiber bending radius introduced by bending on the performance of OAM modes is critical. Thus, we investigate the effect of fiber bending on modal characteristics by an equivalent straight fiber model in view of conformal mapping [25,31]. The guided mode number, maximum bend-induced loss (α), min(Δn_eff) and crosstalk between adjacent MGs |l| = 4, 5, 6, 7 versus fiber bending radius are calculated, as shown in Table 2. One can see that, as the bending radius decreases, the supported mode number would decrease, and maximum bend-induced loss under the circumstance of same mode number would increase. The maximum bend-induced loss usually occurs in the highest-order MG, of which the effective refractive index is closer to the refractive index of cladding and consequently more likely to leak into the cladding. Besides, the maximum α of supported modes are less than 1.86×10⁻⁴ dB/km maintaining in the low level under different bending radii. When the bending radius is larger than 3 cm, the fiber keeps supporting 30 modes with crosstalk between adjacent MGs with |l| = 4, 5, 6, 7 less than -24 dB/km and min(Δn_eff) larger than 1.19×10⁻³.

Table 2. The guided mode number, maximum bend-induced losses (α), min(Δn_eff) and crosstalk between adjacent MGs with |l| = 4, 5, 6, 7 under different fiber bending radius

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Similarly, we discuss the effect of ellipticity on the performance of supported OAM modes at 1550 nm as listed in Table 3. The guided mode number, min(Δn_eff) and crosstalk between adjacent MGs with |l| = 4, 5, 6, 7 under different fiber ellipticity are calculated. The min(Δn_eff) between adjacent low-order MGs (|l| = 0, 1, 2, 3) decreases with the increase of ellipticity, while min(Δn_eff) between adjacent high-order MGs (|l| = 4, 5, 6, 7) increases with the increase of ellipticity and it keeps larger than 1.2 × 10⁻³ under an ellipticity of 5%. In addition, the level of crosstalk is comparable to that of the fiber without disturbance in Table 1. From the above analysis, it can be seen that the designed weakly-guiding graded-index RCF with trench-assisted structure has good tolerance to bending and ellipticity.

Table 3. The guided mode number, min(Δn_eff) and crosstalk between adjacent MGs with |l| = 4, 5, 6, 7 under different fiber ellipticity

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4.5 Wavelength-dependent performance

We further evaluate the wavelength-dependent performance of the designed fiber over the whole C band (1530 nm to 1565 nm). Since the degenerate OAM modes in same MGs have similar n_eff, nonlinearity and dispersions, we only take one of the degenerate OAML/R ±l,1 modes into consideration which is denoted as OAM_l,1. Figure 5(a) shows the calculated mode number and min(Δn_eff) between adjacent high-order MGs (|l| = 4, 5, 6, 7). One can see that the mode number remains at 30 indicating that the designed fiber stably supports up to 7^th order MG over the C band and the min(Δn_eff) between high-order MGs increases as wavelength increasing with minimum value larger than 1.15 × 10³. As analyzed above, for OAM MGs with |l| = 4, 5, 6, 7, the maximum inter-group crosstalk appears between 4th and 5th order MGs and has a maximum value of −25.25 dB at the wavelength ranging from 1530 to 1565 nm, which is displayed in Fig. 5(b). Figs. 5(c)–(f) show the D_λ [22], intra-group DMD, effective mode area (A_eff) [32] and nonlinearity of different modes versus wavelength within the C band. It is noted that the D_λ and intra-group DMD of all the MGs have relatively flat distribution characteristics over the C band, which are within [10.38, 37.62] ps/nm/km and [0.89, 129] ps/km, respectively. The flat low-level D_λ and intra-group DMD are beneficial to weaken the effect of pulse time broadening caused by different frequency components or different modes due to different velocities, showing application potential for small-scale MIMO-DSP assisted 16 modes (MGs with |l| = 4, 5, 6, 7) multiplexing transmission and wavelength division multiplexing transmission. Figs. 5(e) and 5(f) show that the A_eff of all the OAM modes slightly increase with wavelength within (351, 361) µm² over the whole C band and maintains at a relatively large level, while the fiber nonlinearity is inversely proportional to the A_eff, decreasing with wavelength accordingly and sits in (0.26, 0.29) km⁻¹W⁻¹. This means that the power threshold of nonlinear effects is higher, and the designed RCF can transmit higher power signals without causing nonlinear effects, such as self-phase modulation, four-wave mixing, etc.

5. Conclusion

In summary, we design a weakly-guiding trench-assisted graded-index RCF supporting 8 OAM mode groups. For higher order OAM MGs (|l| = 4, 5, 6, 7), the inter-group Δn_eff is larger than 1.18×10⁻³ and the inter-group crosstalk is less than −25 dB/km indicating that the 4 OAM MGs are fully separated. Besides, the maximum intra-group DMD is smaller than 125 ps/km which is beneficial to weaken pulse time broadening caused by different modal velocities and subsequently simplify the complexity of MIMO DSP in optical transmission lines. The effective mode area of each MG is greater than 350 µm² which is able to avoid nonlinear damage in the fiber. Moreover, the designed RCF also has good tolerance to bending and ellipticity, and relatively low fiber attenuation due to its fluorine-doped silica cladding. This kind of fiber design shows great potential to support 16-channel long distance MDM systems with simplified MIMO-DSP by combination of intra-group and inter-group MDM.

Compared with the reported design/fabrications [33–36], the weakly-guiding trench-assisted graded-index RCF is not more complex, on the contrary, it is the common graded-index fiber totally compatible with existing fiber manufacturing processes. In other words, the refractive index profile of the design will not complicate the fiber fabrication process. We have already comprehensively considered the matters in designed fiber that may be encountered in practice in terms of theory, as the design idea is put forward and feasibilities are proved from theory, our next work is to fabricate the fiber and test the properties for wavelength division multiplexing and OAM mode multiplexing. It would be a laudable engineering effort to fabricate the design fiber and prove the high-performance of this kind of fiber experimentally.

Funding

National Key Research and Development Program of China (2018YFB1801803); National Natural Science Foundation of China (62125503); Key R&D Program of Hubei Province of China (2020BAB001, 2021BAA024); Science and Technology Innovation Commission of Shenzhen (JCYJ20200109114018750); Fundamental Research Funds for the Central Universities (2019kfyRCPY037).

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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MG order	Eigenmodes	OAM modes	Δn_eff (×10⁻³)	Inter-group XT (dB/km)	Intra-group DMD (ps/km)
4	$\begin{array}{cc} H E_{51}^{e / o} & E H_{31}^{e / o} \end{array}$	$O A M_{\pm 4, 1}^{L / R}$	1.189	−25.65	16.17
5	$\begin{array}{cc} H E_{61}^{e / o} & E H_{41}^{e / o} \end{array}$	$O A M_{\pm 5, 1}^{L / R}$	1.439	-29.94	39.21
6	$\begin{array}{cc} H E_{71}^{e / o} & E H_{51}^{e / o} \end{array}$	$O A M_{\pm 6, 1}^{L / R}$	1.679	-33.25	74.65
7	$\begin{array}{cc} H E_{81}^{e / o} & E H_{61}^{e / o} \end{array}$	$O A M_{\pm 7, 1}^{L / R}$	-	-	124.09

Bending radius (cm)		1	2	3	5	10	20	50
Mode number		18	26	26	26	30	30	30
Maximum α (dB/km)		1.86e-4	1.95e-4	2.42e-7	1.44e-7	1.94e-5	4.78e-6	2.98e-6
Minimum (Δn_eff)		1.15e-3	1.18e-3	1.19e-3	1.19e-3	1.19e-3	1.19e-3	1.19e-3
XT (dB/km)	(4,5)	−20.55	−23.08	−23.91	−24.60	−25.13	−25.40	−25.56
	(5,6)	−	−27.63	−28.37	−28.98	−29.47	−29.71	−29.86
	(6,7)	−	−	−	−36.06	−32.81	−33.04	−33.18

Ellipticity		0.1%	0.5%	1%	5%	8%	10%	15%
Mode number		30	30	30	30	26	26	26
Min (Δn_eff) between low-order MGs		1.04e-4	1.03e-4	9.77e-5	5.04e-5	2.44e-5	1.15e-5	9.47e-6
Min (Δn_eff) between high-order MGs		1.19e-3	1.20e-3	1.20e-3	1.24e-3	1.28e-3	1.31e-3	1.37e-3
XT (dB/km)	(4,5)	−25.78	−26.23	−26.77	−30.65	−33.26	−34.74	−37.87
	(5,6)	−30.08	−30.51	−31.04	−34.78	−37.27	−38.67	−41.55
	(6,7)	−33.38	−33.79	−34.31	−37.92		−	−

MG order	Eigenmodes	OAM modes	Δn_eff (×10⁻³)	Inter-group XT (dB/km)	Intra-group DMD (ps/km)
4	$\begin{array}{cc} H E_{51}^{e / o} & E H_{31}^{e / o} \end{array}$	$O A M_{\pm 4, 1}^{L / R}$	1.189	−25.65	16.17
5	$\begin{array}{cc} H E_{61}^{e / o} & E H_{41}^{e / o} \end{array}$	$O A M_{\pm 5, 1}^{L / R}$	1.439	-29.94	39.21
6	$\begin{array}{cc} H E_{71}^{e / o} & E H_{51}^{e / o} \end{array}$	$O A M_{\pm 6, 1}^{L / R}$	1.679	-33.25	74.65
7	$\begin{array}{cc} H E_{81}^{e / o} & E H_{61}^{e / o} \end{array}$	$O A M_{\pm 7, 1}^{L / R}$	-	-	124.09

Bending radius (cm)		1	2	3	5	10	20	50
Mode number		18	26	26	26	30	30	30
Maximum α (dB/km)		1.86e-4	1.95e-4	2.42e-7	1.44e-7	1.94e-5	4.78e-6	2.98e-6
Minimum (Δn_eff)		1.15e-3	1.18e-3	1.19e-3	1.19e-3	1.19e-3	1.19e-3	1.19e-3
XT (dB/km)	(4,5)	−20.55	−23.08	−23.91	−24.60	−25.13	−25.40	−25.56
	(5,6)	−	−27.63	−28.37	−28.98	−29.47	−29.71	−29.86
	(6,7)	−	−	−	−36.06	−32.81	−33.04	−33.18

Weakly guiding graded-index ring-core fiber supporting 16-channel long distance mode division multiplexing systems based on OAM modes with low MIMO-DSP complexity

Abstract

1. Introduction