1. INTRODUCTION

The undamped exponential growth of cloud-based traffic, highly centralized data, and mobile services has been stimulating the exploration of expanding the capacity of networks. Miscellaneous multiplexing technologies, such as time-division multiplexing, wavelength-division multiplexing (WDM), and polarization-division multiplexing, have been widely applied to ease the capacity crunch. In view of the fact that existing multiplexing technologies have reached their scalability limits, space-division multiplexing (SDM) has emerged with great potential [1,2]. Few-mode fiber (FMF), as a promising candidate for SDM, has drawn close attention in the fiber-optical communication domain in the recent past [3–5].

Remarkably, six spatial and polarization modes have been successfully transmitted simultaneously, each carrying 40 Gbit/s quadrature-phase-shift-keyed channels over 96 km FMF [6]. An obstacle encountered with FMFs is mode coupling, and thus multiple-input multiple-output (MIMO) signal processing has to be used to withstand the induced cross talk, causing rises in the cost and the complexity of system [7]. One straightforward solution is to eliminate the degeneracy between adjacent eigenmodes by enlarging the effective refractive index differences to $\mathrm{\Delta }{n}_{\mathrm{eff}}>{10}^{-4}$ [8]. From this point of view, a polarization-maintaining (PM)-FMF, having an elliptical core with a step-index profile, was suggested to realize MIMO-free systems [9], and the number of guided PM modes has been increased to eight by utilizing an elliptical ring-core fiber (RCF) [10]. However, in the elliptical core FMFs, the higher-order modes tend to cut off under high ellipticity. As reported in Ref. [11], even with a small ellipticity of 1.5, the odd mode of an ${\mathrm{LP}}_{11}$ mode group reaches cutoff at 1550 nm; in Ref. [10], this problem is alleviated by an FMF with an elliptical ring core, but the two fundamental modes remain degenerate with insufficient $\mathrm{\Delta }{n}_{\mathrm{eff}}$ ($\sim {10}^{-5}$).

In recent years, a type of RCF containing a circularly symmetric ring core with a high-contrast refractive index was utilized for the transmission of orbital angular momentum (OAM) modes, which can split the effective refractive indices ($n_{\mathrm{eff}}$) of transverse electric (TE), transverse magnetic (TM), hybrid (HE or EH) modes and greatly simplify the MIMO signal processing [12,13]. The ring-core structure combined with the multicore fiber technique has been proposed for ultrahigh-density SDM [14]. While the OAM modes in RCF are actually the linear combinations of two degenerate even and odd modes of the HE or EH mode groups [15], the $\pm L$-order OAM modes inevitably couple to each other with anisotropic perturbations [16].

In this paper, we propose a PM PANDA RCF (PRCF) that features the combination of a ring core with two stress-applying rods. The ring-core structure, as in the OAM-mode-supported RCFs, is circularly symmetric with a high refractive index contrast between core and cladding, thus effectively preventing cutoffs of the higher-order modes and splitting the HE, TE, TM, and EH modes. The two stress-applying rods induce birefringence, separating the residual degenerate modes, i.e., the even and odd HE or EH modes [17,18]. The fiber structure parameters are selected to enable support of 10 polarization- and spatial-distribution-maintaining modes, with all the effective refractive index differences between the adjacent modes satisfying $\mathrm{\Delta }{n}_{\mathrm{eff}}>{10}^{-4}$. Broadband characteristics over a wide wavelength range covering the whole $C$ and $L$ bands are also realizable, which indicates compatibility with the mature WDM technique.

2. FIBER PARAMETER SELECTION AND MODAL PROPERTIES

The schematic cross section and refractive index profile of the proposed PM-PRCF is shown in Fig. 1, where the cladding diameter is $W$ and the gap between the ring core and stress-applying rods is $a$. The inner radius, outer radius of the ring core, and radius of the stress-applying rods are $r_1$, $r_2$, and $r_3$, respectively. The cladding is made of silica with refractive index $n_1=1.444$ at 1550 nm, while the ring core is ${\mathrm{GeO}}_2$-doped silica due its index $n_2=1.474$, which corresponds to 0.202 molecular fraction doping of ${\mathrm{GeO}}_2$ in ${\mathrm{SiO}}_2$ [19]. The materials of the stress-applying rods, having index $n_3=1.436$, are silica doped with 0.3 molecular fraction of ${\mathrm{B}}_2{\mathrm{O}}_3$, which has been reported and utilized in the practical fabrication of high-birefringence fibers [20]. The normalized frequency is defined as $V=r_2·2\pi \sqrt{(n_2^2-n_1^2)}/\lambda$, where $\lambda$ is the wavelength and the radius ratio is set to be $\rho =r_1/r_2$. The detailed elastic material parameters used for modeling are listed in Table 1. The elastic coefficients for the ${\mathrm{SiO}}_2$ cladding and the ${\mathrm{GeO}}_2-{\mathrm{SiO}}_2$ ring core are obtained from Ref. [21], while the corresponding ones for the ${\mathrm{B}}_2{\mathrm{O}}_3-{\mathrm{SiO}}_2$ stress-applying rods can be found in Refs. [17,22]. Numerical calculations are done by finite element analysis with the COMSOL software package.

 

Fig. 1. Schematic diagram of the PM-PRCF cross section.

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Table 1. Elastic Parameters Used for Modeling

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In order to determine the structure size of a PM-PRCF that supports 10 eigenmodes (two fundamental modes, four first higher-order modes, and four second higher-order modes) while meeting the requirement of effective refractive index differences $\mathrm{\Delta }{n}_{\mathrm{eff}}>{10}^{-4}$ between adjacent modes, we fix $W=125\mathrm{\mu m}$, $a=1\mathrm{\mu m}$, and $r_3=20\mathrm{\mu m}$ and sweep the two parameters $V$ (from 4 to 6) and $\rho$ (from 0.3 to 0.85) to calculate $\mathrm{\Delta }{n}_{\mathrm{eff}}$ at 1550 nm. The selection of the two parameters $a$ and $r_3$ will be discussed later. Figure 2 shows a colormap of the minimum values of the effective refractive index differences between any two of the 10 supported modes as a function of $\rho$ and $V$. The point $V=4.51$ and $\rho =0.57$ is chosen as the target fiber structure size, with $n_{\mathrm{eff}}$ and $\mathrm{\Delta }{n}_{\mathrm{eff}}$ for all the modes listed in Table 2. As shown in Table 2 (as well as in Fig. 3), the superscripts of the mode names show the intensity patterns. The numbers in the subscripts refer to azimuthal and radial indices, while the $x$ or $y$ indicates the polarization direction of electrical field.

 

Fig. 2. Colormap of minimal $\mathrm{\Delta }{n}_{\mathrm{eff}}$ between adjacent modes as a function of $\rho$ and $V$ at 1550 nm for $W=125\mathrm{\mu m}$, $a=1\mathrm{\mu m}$, $r_3=20\mathrm{\mu m}$.

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Fig. 3. Intensity distributions and electrical field orientations for the 10 eigenmodes (a) without the two stress-applying rods, (b) with the two stress-applying rods.

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Table 2. Effective Refractive Index ($n_{\mathrm{eff}}$), Effective Refractive Index Difference ($\mathrm{\Delta }{n}_{\mathrm{eff}}$) Between Adjacent Modes, and Chromatic Dispersion ($D$) at 1550 nm for $a=1\mathrm{\mu m}$, $r_3=20\mathrm{\mu m}$, $V=4.51$, and $\rho =0.57$

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Fiber eigenmode profiles are significantly affected by the fiber structure. Benefiting from the high-contrast index ring-core structure, high fields and field gradients exist in the ring core [23]. Figure 3(a) shows the vectorial mode intensity distributions at the ring core in the absence of the two stress-applying rods, along with arrows representing the polarization orientations of electrical fields. All the modes show circularly symmetric patterns and curved polarizations (except for ${\mathrm{HE}}_{11x}$ and ${\mathrm{HE}}_{11y}$). With the introduction of the two stress-applying rods, the mode profiles distort from the ring shapes into corresponding linearly polarized (LP) mode intensity distributions, while the polarization directions become horizontal or vertical, as shown in Fig. 3(b).

Pursuant to the birefringence theory, stress anisotropy induced by the stress-applying rods increases the $\mathrm{\Delta }{n}_{\mathrm{eff}}$ between the two adjacent modes oriented orthogonally in the same order [24], with the intensity of the stress birefringence depending on the gap $a$ between the ring core and the stress-applying rods. As stress is applied to the ring core, the effective index separations occur among the mode pairs, accounting for the evolution from vectorial modes to corresponding LP modes. Along with the decrease in $a$, the stress exerted on the ring core is enhanced, resulting in an increase in all the $\mathrm{\Delta }{n}_{\mathrm{eff}}$ values except for the one between ${\mathrm{LP}}_{11y}^{\mathrm{odd}}$ and ${\mathrm{LP}}_{11y}^{\mathrm{even}}$, shown in Fig. 4. Nevertheless, it is still above ${10}^{-4}$ for $a>1\mathrm{\mu m}$. Although the separations between the $n_{\mathrm{eff}}$ values of the even and odd second higher-order modes (${\mathrm{LP}}_{21y}^{\mathrm{odd}}$ and ${\mathrm{LP}}_{21y}^{\mathrm{even}}$, ${\mathrm{LP}}_{21x}^{\mathrm{odd}}$ and ${\mathrm{LP}}_{21x}^{\mathrm{even}}$) are influenced heavily by $a$, the other six modes (two fundamental modes and four first higher-order modes) continue to satisfy $\mathrm{\Delta }{n}_{\mathrm{eff}}>{10}^{-4}$ within the range $1\mathrm{\mu m}>a>5\mathrm{\mu m}$.

 

Fig. 4. Effective refractive index $n_{\mathrm{eff}}$ and effective refractive index difference $\mathrm{\Delta }{n}_{\mathrm{eff}}$ as a function of $a$.

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The radius of the stress-applying rods is closely related to the difficulty of the manufacturing process and the performance of the PM-PRCF. In order to determine the value of $r_3$ to simplify the fabrication process and achieve the target $\mathrm{\Delta }{n}_{\mathrm{eff}}$, we investigate the dependence of $\mathrm{\Delta }{n}_{\mathrm{eff}}$ on $r_3$. The $\mathrm{\Delta }{n}_{\mathrm{eff}}$ values of the adjacent modes are barely affected by the changes in $r_3$, with the exceptions of ${\mathrm{LP}}_{01x}$ and ${\mathrm{LP}}_{01y}$, ${\mathrm{LP}}_{21y}^{\mathrm{even}}$ and ${\mathrm{LP}}_{21x}^{\mathrm{odd}}$, and ${\mathrm{LP}}_{11y}^{\mathrm{even}}$ and ${\mathrm{LP}}_{11x}^{\mathrm{even}}$, as shown in Fig. 5. Though their $\mathrm{\Delta }{n}_{\mathrm{eff}}$ values diminish when $r_3$ decreases, the differences remain $>{10}^{-4}$ within the region of $r_3>8\mathrm{\mu m}$. The designed PM-PRCF presents enhanced tolerance to the variations of $r_3$, and hence it is appropriate to choose a small value for $r_3$ such as 8 μm, for which the fabrication process is simpler.

 

Fig. 5. Effective index difference $\mathrm{\Delta }{n}_{\mathrm{eff}}$ as a function of $r_3$.

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By taking advantage of the high-contrast index ring core, the designed PM-PRCF gains large effective index separations between the modes with different polarization manifestations, namely, ${\mathrm{TE}}_{01}/{\mathrm{TM}}_{01}$ and ${\mathrm{HE}}_{21}$ for the first higher order, ${\mathrm{HE}}_{31}$ and ${\mathrm{EH}}_{11}$ for the second higher order. The introduction of the two stress-applying rods brings birefringence, splitting the remaining degenerate modes, i.e., the even and odd modes in the same order. Due to the unique capability of the PM-PRCF, the 10 modes are all separated from their adjacent modes at the selected fiber structure size in Fig. 2, with the maximum $\mathrm{\Delta }{n}_{\mathrm{eff}}$ as large as $6.41\times {10}^{-3}$ between the ${\mathrm{LP}}_{11y}^{\mathrm{odd}}$ and ${\mathrm{LP}}_{21x}^{\mathrm{even}}$ modes; the minimum $\mathrm{\Delta }{n}_{\mathrm{eff}}$ can still reach $1.29\times {10}^{-4}$ between the ${\mathrm{LP}}_{21y}^{\mathrm{even}}$ and ${\mathrm{LP}}_{21y}^{\mathrm{odd}}$ modes, as seen in Table 2.

3. BROADBAND CHARACTERISTICS

We evaluate the performance of the PM-PRCF over a wide wavelength range from 1500 to 1630 nm, covering the whole $C$ and $L$ bands. The refractive indices of ${\mathrm{SiO}}_2$, ${\mathrm{GeO}}_2-{\mathrm{SiO}}_2$, and ${\mathrm{B}}_2{\mathrm{O}}_3-{\mathrm{SiO}}_2$ at different wavelengths can be found in Refs. [19,20]. Results show the minimal $\mathrm{\Delta }{n}_{\mathrm{eff}}$ for the 10 modes over such a wide range is $1.12\times {10}^{-4}$. The chromatic dispersions ($D$) of all modes are compatible with the values of standard single-mode fiber, except that the maximum value of $D$, $-104\mathrm{ps}/\mathrm{nm}/\mathrm{km}$ for ${\mathrm{LP}}_{21y}^{\mathrm{odd}}$ at 1630 nm, is a little large, as shown in Fig. 6. However, it can be compensated by the established dispersion-compensation techniques [25,26]. The values of $D$ for 10 modes at 1550 nm are given in Table 2.

 

Fig. 6. Chromatic dispersions for the 10 modes.

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As for the fabrication of the designed PM-PRCF, we believe that it can been achieved using well-developed fiber manufacture technologies. The ring-core structures with high refractive index contrast (and more complex structures) have been realized through a modified/plasma chemical vapor deposition process and used for the transmission of OAM modes [27,28]. The stress-applying rods have been widely used in the PANDA fibers to maintain mode polarizations [29–31]. The existing fiber manufacturing technologies provide a solid base for the fabrication of this PM-PRCF.

4. CONCLUSION

In conclusion, we have presented the design of a PM-PRCF supporting 10 polarization- and spatial-distribution-maintaining modes for SDM. The large effective refractive index differences of adjacent modes ($>{10}^{-4}$) allow the maintenance of both the electrical field polarizations and intensity distributions. Broadband performance is demonstrated over a wide wavelength range covering the whole $C$ and $L$ bands with small chromatic dispersions. We are confident that the PM-PRCF will have broad application in MIMO-free SDM combined with WDM for improving optical communication capacity.