## Abstract

An optimized design of 7-ring-core 5-mode-group fiber for mode-group-based dense space-division multiplexing (DSDM) is proposed. It is found that decreasing the refractive index of the center and trench of each ring core can increase the available ring-core thickness and meanwhile suppress the radial higher-order modes. Based on the simulation, a fiber, with a thicker ring core and a relatively low refractive index contrast at the ring-core boundaries, is found of higher mode purity and lower macro-bending-caused mode coupling. In the experiment, the seven cores of the fabricated fiber have low transmission losses around −0.25 dB/km with only a few fluctuations. The three higher-order mode groups (MG_{2,1}, MG_{3,1}, and MG_{4,1}) are verified to be in a weak-coupling state (crosstalk of which are less than −12 dB) over a transmission length of 23 km.

© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. Introduction

Currently, the demand for the data transmission capacity of optical communication networks is increasing rapidly. Space-division multiplexing (SDM) provides a feasible method to further improve the data transmission capacity [1,2]. In fiber optics, SDM consists of two ways, one is core-division multiplexing (CDM), the other one is mode-division multiplexing (MDM). The technique of CDM is based on multi-core fiber. When the crosstalk between fiber cores ($\textrm {XT}_{\textrm {c}}$) is sufficiently small, each core can be regarded as an independent signal channel. Many works have been done for reducing the $\textrm {XT}_{\textrm {c}}$, including the design of heterogeneous and trench-assisted multi-core fibers [3–7]. By optimizing the core structure and arrangement, ultra-low $\textrm {XT}_{\textrm {c}}$ and high core density can be achieved simultaneously [6]. The idea of MDM dates back 1980s [8]. In ideal few-mode or multi-mode fibers, because of mode orthogonality, each mode can be used as an independent signal channel. However, fiber defects introduced by fabrication and external perturbations can break the orthogonal condition and cause crosstalk between fiber modes ($\textrm {XT}_{\textrm {m}}$), which will damage the signal quality. Therefore, solving the problem of crosstalk is essential to the SMD technique.

From the point of view of signal processing, the multiple-input multiple-output (MIMO) technique can be used to compensate for the crosstalk [9]. However, as the number of modes increases, the complexity of MIMO signal processing also increases. In addition, the differential mode group delay (DMGD) causes the time shift of signal in different mode channels at the receiver end. Thus, multiple tapped delay lines need to be used in the receiver for the full MIMO signal processing. To reduce the complexity of MIMO processing with a relatively large mode number, specially designed step-index few-mode fiber (SI-FMF) [10], ring-assisted few-mode fiber (RA-FMF) [11,12],and ring-core fiber (RCF) [13,14] were proposed for MIMO-less mode group transmission. Such characteristic can achieve small crosstalk between mode groups ($\mathrm {XT}_{\mathrm {mg}}$). Although the modes in the same mode group remain strongly coupled because of the degeneracy, their $\mathrm {XT}_{\mathrm {m}}$ can be compensated by relatively simple MIMO signal processing. For the SI-FMF, 4 linearly polarized (LP) mode groups ($\mathrm {LP}_{0,1}$, $\mathrm {LP}_{1,1}$, $\mathrm {LP}_{2,1}$, and $\mathrm {LP}_{0,2}$) were demonstrated that they can propagate with low mode coupling [10]. In 2018, the RA-FMF which can further increase the difference of effective refractive index ($\Delta n_{\mathrm {eff}}$) was reported [11]. With this optimized design, the weakly-coupled transmission of 6 LP mode groups ($\mathrm {LP}_{0,1}$, $\mathrm {LP}_{1,1}$, $\mathrm {LP}_{2,1}$, $\mathrm {LP}_{0,2}$, $\mathrm {LP}_{3,1}$ and $\mathrm {LP}_{1,2}$) was demonstrated [12]. Another way for mode-group-based SDM transmission is using RCF, in which all the radial higher-order modes can be filtered out without damaging the existence of radial first-order modes [15]. Therefore, inter-mode-group (inter-MG) coupling caused by the radial higher-order modes (like $\mathrm {LP}_{0,2}$ and $\mathrm {LP}_{1,2}$) can be eliminated. Orbital angular momentum (OAM) modes have ring intensity profiles that are similar to a ring-core structure. These circular symmetry intensity profiles are beneficial to achieve ultra-low differential mode gain of fiber amplifier [16]. In addition, the maximum degeneracy of a higher-order OAM mode group is 4, thus only $4\times 4$ MIMO is needed to compensate the $\textrm {XT}_{\textrm {m}}$ of OAM modes in the same mode group. The SDM transmission using OAM modes in 10-km [13] and 100-km [14] RCFs with the assistance of $4\times 4$ MIMO have been demonstrated, respectively. Thus, the OAM mode-group-based SDM scheme is an important candidate for increasing data transmission capacity.

Dense space-division multiplexing (DSDM), based on multi-core few-mode fiber, can greatly increase optical communication capacity. An experimental result of $255$ $\mathrm {Tbit/s}$, achieved by multiplexing two LP modes in a 7-core few-mode fiber, has been reported [17]. More recently, a 39-core 3-mode fiber was used for DSDM data transmission [18,19]. The cores of the fiber have a graded-index profile, and the three LP modes in each core are strongly coupled. Taking into account the two polarization states of each LP mode, $6\times 6$ MIMO signal processing was used at the receiver end. Combining the multiplexing of cores and modes, this fiber can support 228 spatial channels. In this 13-km long fiber, a $10.66\ \mathrm {Pb/s}$ data transmission was achieved. However, the design, fabrication, and test of multi-ring-core few-mode fiber for OAM mode-group-based DSDM transmission have not been reported.

In this paper, we first report the design of a 7-ring-core fiber, of which each ring core can support a fundamental mode group and four higher-order OAM mode groups in the C+L band, for the mode-group-based DSDM data transmission. The modes in the same mode group are strongly coupled, while the mode groups are weakly coupled. Thus, only a relatively simple MIMO signal processing is required to compensate the $\mathrm {XT}_{\mathrm {m}}$ between modes in the same mode group. As the $n_{\mathrm {eff}}$ of mode groups are easily separated, such a fiber does not need ultrahigh refractive index contrast. It is found that applying negative doping at the center and trench of a ring core can help to suppress the radial higher-order modes with a relatively large ring-core thickness. Hence, the appropriate ring-core thickness and negative doping ratio guarantee the low mode coupling between mode groups and high mode purity, so that, the robustness of the fiber can be ensured. Based on the fabricated fiber, low transmission loss, and low $\textrm {XT}_{\textrm {mg}}$ between mode groups are demonstrated in the experiment. The contents of the paper are organized as follows. In section 2, we introduce the detailed design method of the ring-core structure. And, the $\textrm {XT}_{\textrm {c}}$ between the adjacent ring cores are discussed based on the average coupled-power theory. The fabricated fiber and measured results are given in section 3. Section 4 is the conclusion.

## 2. Fiber design

The schematic refractive index profile of half of the RCF is shown in Fig. 1(a). It consists of a center, a ring core, a middle cladding, a trench, and an outer cladding. The ring core has a high refractive index $n_{\mathrm {co}}$, and the inner and outer radii of the ring core are $r_{1}$ and $r_{2}$, respectively. The refractive index of the center is $n_{\mathrm {ce}}$, which is lower than the refractive index of pure silica. The trench layer also has a relatively low refractive index $n_{\mathrm {tr}}$, it can increase the modal confinement, which is widely used to reduce the $\textrm {XT}_{\textrm {c}}$ in multi-core fibers [4]. Inner and outer radii of the trench are $r_{3}$ and $r_{4}$, respectively. And, the middle and outer cladding are pure fused silica, their refractive index ($n_{\mathrm {cl}}$) is calculated by the Sellmeier formula [20].

#### 2.1 Mode purity

OAM modes and vector modes are equivalent orthogonal mode bases of optical fiber. The vector modes in a RCF can be obtained by solving the Helmholtz equations [15]. And, OAM modes can be expressed by the superposition of the even and odd vector modes with a $\pm \pi /2$ phase difference. The electric and magnetic components are expressed as

Based on the continuous boundary conditions of the tangential optical fields, by assuming that $n_{\mathrm {eff}}\rightarrow n_{\mathrm {cl}}$, we can calculate the cutoff conditions of each mode group. The design regions for exactly supporting five mode groups (a fundamental mode group and four higher-order OAM mode groups) with filtering out all the radial higher-order modes in the whole C+L band are shown in Fig. 1(b). The bottom solid boundary and right dashed boundary of each design region are the cutoff curves of the first radial higher-order mode group $\mathrm {MG}_{0,2}$ (contains modes $\mathrm {OAM}^{\pm }_{0,2}$) and the fifth azimuthal higher-order mode group $\mathrm {MG}_{5,1}$ (contains modes $\mathrm {OAM}^{\pm }_{\pm 5,1}$ and $\mathrm {OAM}^{\mp }_{\pm 5,1}$), respectively, at the wavelength of 1530 nm. And, the left doted boundary is the cutoff line of the fourth azimuthal higher-order mode group $\mathrm {MG}_{4,1}$ (contains modes $\mathrm {OAM}^{\pm }_{\pm 4,1}$ and $\mathrm {OAM}^{\mp }_{\pm 4,1}$) at the wavelength of 1625 nm. The relative refractive index difference of the doped area with the cladding is defined as $\Delta _{\mathrm {area}} = (n_{\mathrm {area}}-n_{\mathrm {cl}})/n_{\mathrm {area}}\times 100\%$. In Fig. 1(b), $\Delta _{\mathrm {co}}$ is fixed at $0.6\%$, while $\Delta _{\mathrm {ce}}$ and $\Delta _{\mathrm {tr}}$ are changed. It can be seen that the cutoff curve of $\mathrm {MG}_{0,2}$ shifts towards the smaller value of $r_{1}/r_{2}$ with the decrease of $\Delta _{\mathrm {ce}}$ and $\Delta _{\mathrm {tr}}$, which indicates that the negative doping of the center and trench helps to suppress the radial higher-order mode even with thicker ring core. It should be noted that the black curves represent the case of ring core with a hollow center, of which the refractive index of the central area is 1.

#### 2.2 Macro bending caused mode-group coupling

Fiber bending can break the orthogonality of ideal fiber modes and lead to mode coupling. In an ideal optical fiber, the circularly symmetrical refractive index profile is $n_{0}(x,y)=n_{0}(r)$. According to coupled-mode theory [23], the mode coupling coefficient $c_{mn}$ between $\mathrm {OAM}_{m,1}$ and $\mathrm {OAM}_{n,1}$ can be expressed by the overlap integral in the cross section of optical fiber that

The function $\cos (\phi )$ in Eq. (6) indicates that the macro bending mainly induces coupling between adjacent mode groups $\mathrm {MG}_{l_{1},1}$ and $\mathrm {MG}_{l_{2},1}$ ($|l_{1}-l_{2}|=1$).

The design points A, B, C, and D in Fig. 1(b) are chosen to compare their ability to resist mode coupling. Figure 2(a) shows the $n_{\mathrm {eff}}$ of modes for the four design points. The $n_{\mathrm {eff}}$ of $\mathrm {MG}_{4,1}$ does not change much with the change of design point, because it is determined by the cladding refractive index. However, the $n_{\mathrm {eff}}$ of the other lower-order mode groups increase by more than $2\times 10^{-4}$ for B, C, and D compared with A. Hence, B, C, and D provide larger $\Delta n_{\mathrm {eff}}$ between mode groups, which is conducive to preventing the coupling between mode groups. The normalized dominant coupling coefficients caused by macro bending (when $R_{\mathrm {b}} = 8$ cm) are shown in Fig. 2(b). The design point A, which has the thinnest ring core, is most easily affected by macro bending. Design points B and C have a relatively thicker ring core, and the refractive index contrasts at the inner ring core boundary of them are not as high as that of design point D, thus, their coupling coefficients are smaller than those of design point D.

In summary, the thickness of the ring core and the refractive index contrast at the ring core boundaries jointly influence the coupling coefficients and purity of modes. The negative doping at the center and trench can effectively extend the available ring-core thickness, hence, can decrease the coupling coefficients under macro bending. On the other hand, a high refractive index contrast at the boundaries of the ring core, such as the case of ring core with a hollow center, can severely damage the mode purity. Although the ring core is thickest in the case of hollow center, the macro bending can cause a relatively high coupling coefficient. Therefore, the design points B and C are better than A and D.

#### 2.3 Selection of core refractive index

According to the above discussion, points B and C in Fig. 1(b) have good properties of achieving high mode purity and low coupling coefficient. However, the high negative doping ratio of the point C is rarely used for the fabrication of fiber preform, thus we chose the point B, of which $\Delta _{\mathrm {ce}}=\Delta _{\mathrm {tr}} = -0.4\%$, for the design. We further discuss the effects of changing the core refractive index on $\Delta n_{\mathrm {eff}}$, DMGD, mode purity, and effective mode area (EMA). The cutoff curves of $\mathrm {MG}_{0,2}$, $\mathrm {MG}_{4,1}$ and $\mathrm {MG}_{5,1}$ at the wavelengths of 1530 and 1625 nm are shown in Fig. 3. When the selected point is in the bottom left region to the cutoff line, the corresponding mode group is cutoff. The colored area is the target design region. To reduce the $\textrm {XT}_{\textrm {mg}}$, an effective method is to improve the inter-MG $\Delta n_{\mathrm {eff}}$. In a RCF, the lowest difference of $n_{\mathrm {eff}}$ ($\Delta n^{0/1}_{\mathrm {eff}}$) exists between the fundamental mode group $\mathrm {MG}_{0,1}$ and the first higher-order mode group $\mathrm {MG}_{1,1}$. The variation of the $\Delta n^{0/1}_{\mathrm {eff}}$ is illustrated in Fig. 3(a). With the decrease of $r_{2}$ and increase of $\Delta _{\mathrm {co}}$ the $\Delta n^{0/1}_{\mathrm {eff}}$ increases constantly from $1.8\times 10^{-4}$ to $7\times 10^{-4}$. Thus, a larger ring-core refractive index and a smaller ring-core size are better for improving the separation of $n_{\mathrm {eff}}$ of mode groups. On the other hand, the DMGD between modes in the same mode group should be minimized to reduce the complexity of MIMO signal processing. The DMGD between the near-degenerate modes in the highest order mode group $\mathrm {MG}_{4,1}$ is discussed as an example, we show the DMGD between $\mathrm {OAM}^{\pm }_{\pm 4,1}$ and $\mathrm {OAM}^{\mp }_{\pm 4,1}$ in Fig. 3(b). In the design region, the DMGD decreases from 290 to 27 ps/km when $r_{2}$ grows from 8 to 15 $\mu \mathrm {m}$. The variations of mode purity and EMA of $\mathrm {OAM}_{0,1}$ are shown in Fig. 3(c) and 3(d), respectively. Both of them increase with the growth of $r_{2}$ and the decrease of $\Delta _{\mathrm {co}}$. A higher purity can prevent the intrinsic $\textrm {XT}_{\textrm {mg}}$, and a larger EMA helps to reduce the nonlinear effects. Taking an overall consideration of $\Delta n_{\mathrm {eff}}$, DMGD, pruity and EMA, the design point is finally set at $r_{2} = 11\ \mu \mathrm {m}$ and $\Delta _{\mathrm {co}}=0.6\%$ (noted by the gray spot in Fig. 3). The simulation results at the design point are $\Delta n_{\mathrm {eff}}^{0/1}\simeq 3.54\times 10^{-4}$, DMGD in $\mathrm {MG}_{4,1}$ $\simeq$ 100 ps/km, purity of $\mathrm {OAM^{\pm }_{0,1}}\simeq 99.995 \%$, and EMA of $\mathrm {OAM^{\pm }_{0,1}} \simeq 315\ \mu \mathrm {m}^{2}$.

#### 2.4 Multi-core arrangement

In a weakly-coupled multi-core fiber, the $\textrm {XT}_{\textrm {c}}$ should be suppressed. Typically, for a multi-core single-mode fiber, the core-to-core distance is around 40 $\mu \mathrm {m}$ to keep the $\textrm {XT}_{\textrm {c}}$ lower than $-30$ dB with the transmission distance longer than 10 km [25]. For our design, the outer radius $r_{2}$ of the ring core is $11\ \mu \mathrm {m}$, thus, the core-to-core distance should be $>51\ \mu \mathrm {m}$ to make the gap between the outer boundary of the adjacent ring cores larger than $40\ \mu \mathrm {m}$. In the condition of weak coupling, the adjacent cores can be regarded as perturbations of each other. Thus, the inter-core coupling coefficient between mode $m$ in core $i$ and the mode $n$ in core $j$ can be expressed by the overlap integral as [23]

On the other hand, the core-to-core distance is restricted by the cladding diameter. The cladding diameter should not exceed $250\ \mu \mathrm {m}$ to maintain the mechanical reliability of a fiber in a lifetime of 20 years [26]. In addition, the outer cladding thickness (the gap between the outer boundary of the side ring core and the coating) should be larger than $40\ \mu \mathrm {m}$ to avoid the excess loss on the side ring cores [27]. To balance the requirement of low $\rm {XT}_{\rm {c}}$, relatively small cladding diameter, and relatively large outer cladding thickness, we choose that the core-to-core distance is $53\ \mu \mathrm {m}$ and the cladding diameter is $220\ \mu \mathrm {m}$, which reserves an outer cladding thickness of $46\ \mu \mathrm {m}$. For different modes, the coupling coefficients have different values. To shows its variation trends, we normalize $c^{(i)(j)}_{mn}$ by its maximum value. When the core-to-core distance is $53\ \mu \mathrm {m}$, the normalized coupling coefficient between each couple of modes in two adjacent cores is shown in Fig. 4(a), it shows that the maximum coupling happens between the highest-order mode groups in the two adjacent cores, which is because of that the highest-order modes have the maximum EMA. To obtain a simulation result approaching the practical condition, the fiber micro deviations and the effects of fiber twisting and macro bending should be taken into account. An exponential autocorrelation function is adopted for describing the micro deviations, and, based on coupled-power theory, the power coupling coefficient has the expression [28]

Then, the $\textrm {XT}_{\textrm {c}}$ at a certain distance $z$ can be obtained by

According to the results in Fig. 4(a), the maximum coupling coefficient exists for $\mathrm {OAM}^{+}_{+4,1}$ and $\mathrm {OAM}^{+}_{-4,1}$. Their $\textrm {XT}_{\textrm {c}}$ in the condition of different bending radius ($R_{\mathrm {b}}$) and autocorrelation length $L_{\mathrm {c}}$ is shown in Fig. 4(b). Because that the $\mathrm {OAM}^{+}_{+4,1}$ and $\mathrm {OAM}^{+}_{-4,1}$ have an intrinsic separation of $n_{\mathrm {eff}}$, there is a phase-matching region (high $\mathrm {XT}_{\mathrm {c}}$) in Fig. 4(b), which represents that the intrinsic separation of $n_{\mathrm {eff}}$ is compensated by the bending-induced $\Delta n_{\mathrm {eff}}$. When $3\ \mathrm {m}<R_{\mathrm {b}}< 10\ \mathrm {m}$, the $\textrm {XT}_{\textrm {c}}$ is larger than $-40\ \mathrm {dB/km}$, which is because of phase matching. And, when $R_{\mathrm {b}}< 3\ \mathrm {m}$, the $\textrm {XT}_{\textrm {c}}$ can keep lower than $-40\ \mathrm {dB/km}$. Therefore, the fiber with the core-to-core distance of $53\ \mu \mathrm {m}$ meets the requirement of low $\textrm {XT}_{\textrm {c}}$ .

## 3. Fabrication and test

The designed fiber was fabricated by the YOFC Company. From the scanning electron microscope (SEM) image shown in Fig. 5(a), the measured core-to-core distance is 53 $\mu \mathrm {m}$ which agrees well with the design. The measured profile of the relative index difference of core $\#$3 is illustrated in Fig. 5(b). Although there are some deviations from the step refractive index, the overall profile can meet the design. Using the measured refractive profile, we recalculated the $n_{\mathrm {eff}}$ by finite element method with COMSOL. The results at the wavelengths of 1545, 1550, and 1555 nm are shown in Fig. 5(c). It indicates that the fiber can support five mode groups and all the radial higher-order modes are forbidden. For the fabricated fiber, because of that, the relative refractive index difference of the cladding with the pure fused silica is around $-0.11\%$, the $n_{\mathrm {eff}}$ of $\mathrm {MG}_{4,1}$ is lower than the designed value. But, it will not cause high transmission loss. For the three higher-order mode groups ($\mathrm {MG}_{2,1}$, $\mathrm {MG}_{3,1}$, and $\mathrm {MG}_{4,1}$), the separations of $n_{\mathrm {eff}}$ between them are relatively large, thus they are weakly coupled with each other. Since their inter-MG $\mathrm {XT}_{\mathrm {mg}}$ is much weaker than the $\mathrm {XT}_{\mathrm {m}}$ of modes in the same mode group, only a $4\times 4$ MIMO is needed for compensating the $\mathrm {XT}_{\mathrm {m}}$. However, the $\Delta n_{\mathrm {eff}}$ among the two lower-order mode groups ($\mathrm {MG}_{0,1}$ and $\mathrm {MG}_{1,1}$) are relatively small, thus, a $6\times 6$ MIMO is required for compensating their $\mathrm {XT}_{\mathrm {m}}$. Then, the corresponding DMGD of each mode relative to the fundamental mode is shown in Fig. 5(d). For the modes belonging to the same mode group, the maximum DMGD exists between $\mathrm {OAM}^{\pm }_{\pm 3,1}$ and $\mathrm {OAM}^{\mp }_{\pm 3,1}$ in $\mathrm {MG}_{3,1}$, which is about 9 times of the design value. It indicates that the refractive index profile deformation has a strong effect on the DMGD.

#### 3.1 Measurement of loss

The transmission losses of modes in the seven cores are measured by using an optical time-domain reflectometer (OTDR). The measured change of relative power in core $\#$1 with the transmission length is shown in Fig. 6(a). The tested fiber has a length of 14.8 km. The signal between the two peaks is originated from Rayleigh scattering. A linear range of the signal can be found between the two peaks. The first sample point is at the beginning of the linear range with a fiber length $L_{1}$ and relative power of $P_{1}$. The second sample point is chosen at the rising edge of the second peak with a fiber length $L_{2}$ and relative power of $P_{2}$. Then the fiber loss coefficient can be calculated by

The measured transmission losses of all mode groups in the seven ring cores are shown in Fig. 6(b). In an overall trend, the higher-order mode groups have a bit larger loss than the fundamental mode group. The lowest loss is 0.241 dB/km found for $\mathrm {MG}_{1,1}$ in core 4. And, the highest loss is 0.257 dB/km found for $\mathrm {MG}_{4,1}$ in core 5. There is only a little deviation of loss among the seven cores, which indicates that the seven cores have a good consistency.

#### 3.2 Measurement of mode-group crosstalk

To verify the law of the $\textrm {XT}_{\textrm {mg}}$ of the above-mentioned fiber, we use the method of power measurement to characterize the $\textrm {XT}_{\textrm {mg}}$. The experimental setups are shown in Fig. 7. The beam emitted by the laser is amplified by an erbium-doped fiber amplifier (EDFA). After collimation, the light beam is converted to OAM beams of $l = -1$, $-2$, $-3$, $-4$, in turn, by vortex phase plates (VPPs). The excited OAM beam is coupled into the 23-km RCF through the objective lens (OJ.). The output beam from the RCF is converted into linear polarization by a linear polarizer. Then the OAM mode is demodulated with a spatial light modulator (SLM) and collimated into a single-mode fiber (SMF) for power measurement. Especially, a flip mirror (FM) is used to facilitate the observation of the demodulated light pattern with a charge-coupled device (CCD).

At the input end, VPPs are used to generate OAM modes having the topological charge of $-1$, $-2$, $-3$, and $-4$, in turn. At the output end, the SLM is loaded with different orders of phase holograms to demodulate and get different demodulated mode patterns as shown in Fig. 8. The patterns in the left column are the output modes from the tested fiber without the demodulation by SLM. To measure the modal components of the output beam, the OAM topological charge of the SLM hologram is tuned from $-4$ to $4$. The mode component has an opposite OAM topological charge with the hologram will be modulated into a Gaussian beam. When the fiber output is $\mathrm {MG}_{1,1}$, the SLM hologram with an OAM topological charge of $1$ can recover a Gaussian-like intensity pattern as shown in Fig. 8($\mathrm {a}_{1+}$). It indicates that most of the energy is on the OAM modes has a topological charge of $-1$. For $\mathrm {MG}_{2,1}$, a clear recovered Gaussian-like pattern exists in Fig. 8($\mathrm {b}_{2+}$), so the OAM modes of $-2$ topological charge has the highest energy. For $\mathrm {MG}_{3,1}$ and $\mathrm {MG}_{4,1}$, Gaussian-like bright spots can be found in Fig. 8($\mathrm {c}_{3+}$) and ($\mathrm {d}_{4+}$), respectively. It illustrates that the energy of output $\mathrm {MG}_{3,1}$ is mainly carried by $\mathrm {OAM}_{-3,1}$, and the energy of output $\mathrm {MG}_{4,1}$ is mainly carried by $\mathrm {OAM}_{-4,1}$.

For the $\textrm {XT}_{\textrm {mg}}$ measurement, considering that the side ring core is easier influenced by fiber bending than the center ring core, a side ring core (core $\#$3) is tested as an example. In the experiment, we measured the $\textrm {XT}_{\textrm {mg}}$ over the entire transmission system includes the $\textrm {XT}_{\textrm {mg}}$ from the beam coupling end, the RCF, and the mode demodulating end. In theory, for each demodulated pattern, only the center Gaussian pattern represents the energy on the mode of the corresponding opposite topological charge. The center Gaussian-like beam is coupled into the SMF by the collimator (Col.) and measured by the power meter (PM). The measurement of $\textrm {XT}_{\textrm {mg}}$ is conducted by fixing the input mode group and changing the topological charge of the SLM hologram. The received power of each mode group is the sum of the power for the corresponding positive and negative topological charge. The measured $\textrm {XT}_{\textrm {mg}}$ are summarized in Table 1. On the general trend, large $\textrm {XT}_{\textrm {mg}}$ happens between adjacent mode groups, which can be attributed to two reasons, one is that the adjacent mode groups have small $\Delta n_{\mathrm {eff}}$, the other one is that macro bending mainly causes mode coupling between adjacent mode groups. In addition, two large values of $\mathrm {XT}_{\mathrm {mg}}$ are found for inputting $\mathrm {MG}_{1,1}$ with outputting $\mathrm {MG}_{3,1}$ and inputting $\mathrm {MG}_{4,1}$ with outputting $\mathrm {MG}_{2,1}$, they may caused by fiber micro deformations like core ellipticity or birefringence [29,30]. The $\textrm {XT}_{\textrm {mg}}$ among the mode groups $\mathrm {MG}_{2,1}$, $\mathrm {MG}_{3,1}$, and $\mathrm {MG}_{4,1}$ over the 23-km long fiber remian below $-12$ dB, and such values were demonstrated to be suitable for the transmission of 16-GBaud QPSK signals [14]. On the other hand, for the $\mathrm {MG}_{0,1}$ and $\mathrm {MG}_{1,1}$ which have the hightest $\mathrm {XT}_{\mathrm {mg}}$ of $-9.11$ dB, a $6\times 6$ MIMO signal processing can be applied for compensating their $\mathrm {XT}_{\mathrm {mg}}$. Because of the lack of fan-in fan-out device, the $\mathrm {XT}_{\mathrm {c}}$ between adjacent cores were not tested in this stage. According to the simulation result, when the bending radius is about 14 cm and the core-to-core distance is more than 50 $\mu \mathrm {m}$, the influence of $\mathrm {XT}_{\mathrm {c}}$ on the signal should be much less than $\mathrm {XT}_{\mathrm {mg}}$.

## 4. Conclusions

A design of 7-ring-core 5-mode-group fiber was proposed for mode-group-based DSDM transmission. The negative doping at the center of the trench-assisted ring core was firstly investigated for resisting the mode-group coupling caused by macro fiber bending. The fabricated fiber has a refractive index profile that agrees well with the design. The modes in the seven cores have low transmission losses, ranging from 0.241 to 0.257 dB/km. The measured $\mathrm {XT}_{\mathrm {mg}}$ among the $\mathrm {MG}_{2,1}$, $\mathrm {MG}_{3,1}$, and $\mathrm {MG}_{4,1}$ are lower than $-12$ dB in the 23-km long fiber, which are suitable for transmitting 16-GBaud QPSK signals [14]. Although the $\mathrm {XT}_{\mathrm {mg}}$ between $\mathrm {MG}_{0,1}$ and $\mathrm {MG}_{1,1}$ is $-9.11$ dB, it can be compensated by $6\times 6$ MIMO signal processing. The calculated $\textrm {XT}_{\textrm {c}}$ is lower than $-40$ dB/km when the bending radius is smaller than 3 m. This fiber can support 35 mode groups for DSDM transmission. With the help of $4\times 4$ and $6\times 6$ MIMO techniques, the available spatial channels are supposed to be 126. Using the proposed design rules with improving the fiber fabrication process, multi-ring-core few-modes fibers with lower loss and better stability of mode groups can be made in the future.

## Funding

National Key Research and Development Program of China (2019YFA0706300); National Natural Science Foundation of China (U1701661, U2001601); Local Innovative and Research Teams Project of Guangdong Pearl River Talents Program (2017BT01X121); Guangzhou Basic and Applied Basic Research Foundation (202002030327); Key-Area Research and Development Program of Guangdong Province (2020B0101080002).

## Acknowledgments

Thank Mr. Xiong Wu for his valuable discussion and advice.

## 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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