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Fully degeneracy-lifted bow-tie elliptical ring-core multi-mode fiber

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Abstract

We present a bow-tie elliptical ring-core multi-mode fiber (BT-ERC-MMF) that features an elliptical ring-core structure and two symmetrical bow-tie stress-applying parts (SAPs). This special fiber design breaking geometry and stress symmetry fully separates the two- or four-fold degenerate modes in traditional circular symmetry fiber with strong mode coupling during mode-division multiplexing (MDM) transmission. The designed fiber is able to support 53 fully degeneracy-lifted eigenmodes with minimum effective index difference between adjacent modes larger than 1.59 × 10−4 at 1550 nm, facilitating potential fiber eigenmode multiplexing transmission without using multiple-input multiple-output digital signal processing (MIMO-DSP) technique. The effect of bending on the designed fiber is investigated based on conformal mapping. Broadband performance including effective modal index (neff), effective index difference (Δneff), effective mode area (Aeff), nonlinearity and chromatic dispersion (D) is also comprehensively studied over the whole C band ranging from 1530 to 1565 nm. The designed fiber targets emerging applications in low-crosstalk direct fiber eigenmode-division multiplexing combined with the mature wavelength-division multiplexing (WDM) technique to increase transmission capacity and spectral efficiency.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

As the traditional available physical dimensions to orthogonally multiplex data have been almost exhausted, space-division multiplexing (SDM) techniques exploiting the space dimension have attracted great interest [1, 2]. Recently, SDM adopting the linearly polarized (LP) modes or orbital angular momentum (OAM) modes in few-mode fiber (FMF), multi-mode fiber (MMF) and vortex fiber has been widely studied in fiber-optic transmission systems to increase transmission capacity and spectral efficiency [3–12]. However, these spatial modes in circular symmetry fiber waveguides are two- or four-fold degenerate, resulting in inter-mode crosstalk during propagation. Hence, with the increase of the multiplexed mode number, multiple-input multiple-output digital signal processing (MIMO-DSP) technique is required at the receiver which introduces additional cost both in complexity and power consumption. An effective approach to obtain negligible modal crosstalk, so as to simplify or eliminate MIMO-DSP, is to lift (i.e. separate) the adjacent fiber eigenmodes with effective index difference (Δneff) values larger than 1 × 10−4, which is also the typical value of birefringence in single-mode polarization-maintaining fibers (PMFs) [13].

Remarkably, ring fibers featuring a high-contrast-index ring-core have proven to be promising to transmit multiple OAM modes stably [5, 14]. In this type of fiber, the difference of effective refractive indices of fiber eigenmodes (HE, EH, TE and TM modes) are split to as large as 1 × 10−4 which benefits negligible modal crosstalk. Note that each HE and EH mode still contains two degenerate even and odd modes, and the two OAM modes synthesized by degenerate even and odd eigenmodes would couple to each other under external perturbations. Furthermore, considering that LP modes and OAM modes can be synthetized by fiber eigenmodes which are orthogonal [15], one can use fiber eigenmodes as an alternative spatial mode basis-set for direct eigenmode-division multiplexing transmission [16]. So far, there have been many research efforts on designing and fabricating the PMFs, including elliptical core FMF [17–19], elliptical ring-core fiber [20, 21], PANDA ring-core fiber [22] and quasi-elliptical arranged supermode fiber [23], to support multiple eigenmodes with enhanced mode spacing for potential MIMO-free eigenmode-division multiplexing transmission. Actually, in the elliptical core FMF supporting three spatial modes, the LP modes are still two-fold degenerate [17–19]. Whereas in the elliptical ring-core fiber supporting ten eigenmodes, the two fundamental modes remain degenerate with insufficient Δneff (∼10−5) [20, 21]. As for the PANDA ring-core fiber and quasi-elliptical supermode fiber, the mode number is limited to 10 [22] and 20 [23]. In this scenario, it would be meaningful to design a fiber with tens of fully lifted eigenmodes which can be used for low-crosstalk direct fiber eigenmode-division multiplexing.

In this paper, we present a bow-tie elliptical ring-core multi-mode fiber (BT-ERC-MMF) design with an elliptical ring-core structure and two symmetrical bow-tie stress-applying parts (SAPs). By combining the ring-core structure and the elliptical core induced geometric birefringence and bow-tie SAPs induced stress birefringence, the designed fiber can support 53 eigenmodes with minimum Δneff between adjacent modes larger than 1.59 × 10−4 at 1550 nm. The stress-optical coupling effects on the fiber eigenmodes are simulated and analyzed by using the full-vector finite-element mode solver. The effect of bending on the designed fiber is investigated by using equivalent straight fiber models for a curved fiber based on conformal mapping. Also, the effective modal index (neff), effective index difference (Δneff), effective mode area (Aeff), nonlinearity and chromatic dispersion (D) for each eigenmode over the whole C band are discussed.

2. Concept

Figures 1(a) and 1(b) show the schematic cross-section and refractive index profile of the designed BT-ERC-MMF, which is composed of two bow-tie (also called sector ring) B2O3-doped-silica SAPs in the pure-silica cladding and symmetrically placed beside a GeO2-doped-silica elliptical ring-core. The semiminor and semimajor axis of the inner ellipse are ax and ay, respectively, with ellipticity denoted as e = ay /ax. The outer ellipse has the semiminor and semimajor axis of bx and by, and the elliptical ring width is denoted as d1 = bx -ax. The two bow-tie SAPs have the inner and outer radii of r1 and r2, and the sector angle is θ. The gap between the outer ellipse and SAPs is d2. The fiber cladding radius b is fixed at 62.5 µm. The relative refractive index difference between the fiber elliptical ring-core (n1) and cladding (n2) is denoted as Δ = (n1- n2)/n2. According to the hybrid Sellmeier equation describing the refractive index dependence on wavelength and concentration (in Mol%) of dopants, we can theoretically get the mole fraction of GeO2 and B2O3 in elliptical ring-core and SAPs which are denoted as n and m [24, 25]. Thermal expansion coefficients of pure SiO2, GeO2 and B2O3 are 5.4 × 10−7 (1/K), 7 × 10−6 (1/K) and 10 × 10−6 (1/K), respectively. Specifically, thermal expansion coefficient (α) of a doped material can be expressed by a role of mixture model shown below [26]

α=(1m)α0+mα1
where α0 and α1 are thermal expanding coefficient of the two kinds of dopants, 1-m and m denote the mole percentage of each dopant. Moreover, other elastic material parameters of fiber material to be used in modeling and simulation are also obtained from Ref [26], including Young’s modulus (E), Poisson’s ratio, first and second stress optical coefficient (B1, B2), operating and reference temperature.

 figure: Fig. 1

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

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3. Fiber parameter selection

To determine the structure size and doping concentrations of a BT-ERC-MMF supporting tens of eigenmodes with Δneff > 10−4 between any two adjacent modes, we sweep eight parameters, including Δ (from 1% to 2.5%), ax (from 1 µm to 5 µm), e (from 1 to 2), d1 (from 1 µm to 5 µm) and m (from 10% to 30%), θ (from 60° to 150°), d2 (from 0.5 µm to 4 µm), r2 (from 30 µm to 60 µm) to calculate minimum Δneff at 1550 nm. Table 1 clearly lists the parameters swept to determine the fiber structure. Note that larger Δ and m enable more fully lifted eigenmodes. Meanwhile, considering of the traditional fiber manufacture facility restriction, we keep Δ ≤ 2.5% and m ≤ 30%. In detail, we firstly fix the structure of two bow-tie SAPs with m = 30% which has been reported in the practical fabrication of high-birefringence fibers [27], θ = 120°, d2 = 1 µm, r2 = 50 µm, and sweep the other four parameters. Table 2 lists the optimal design supporting the most number of fully degeneracy-lifted eigenmodes with minimum Δneff >10−4 under different Δ. We can clearly see that the maximum mode number is 53 with the minimum Δneff equal to 1.59 × 10−4 when Δ is set to be 2.5% and the other three parameters, ax, d1 and e, are equal to 1.4 µm, 4.2 µm and 2, respectively. Furthermore, Table 3 depicts the optimal design supporting the most number of fully degeneracy-lifted eigenmodes with minimum Δneff >10−4 under different ellipticity when fixing Δ = 2.5%. With the increase of the ellipticity of fiber ring-core from 1.1 to 2, the available fully degeneracy-lifted eigenmodes number would increase from 20 to 53. However, considering the traditional fiber manufacture facility restriction, it is preferred that the ellipticity should be as small as possible, thus we keep it no more than 2. It is believed that the design is able to support more fully lifted eigenmodes once the manufacture technology could be further improved. Additionally, we display the minimum Δneff values between adjacent modes and corresponding mode number as a function of d1 and ax when fixing Δ = 2.5% and e = 2 in Figs. 2(a) and 2(b). To meet the requirement of minimum Δneff >10−4 while supporting eigenmodes as many as possible, we choose the point d1 = 4.2 µm and ax = 1.4 µm in the black dotted circle as the final design. Remarkably, within the region of 1.38 µm ≤ ax ≤ 1.43 µm and 4.0 µm ≤ d1 ≤ 4.36 µm, the fiber keeps supporting more than 49 eigenmodes with minimum Δneff larger than 1 × 10−4. That is, the tolerances of ax and d1 are 3.6% and 8.6%, respectively.

Tables Icon

Table 1. Parameters Swept to Determine the BT-ERC-MMF Structure

Tables Icon

Table 2. Optimal Design Supporting the Most Number of Fully Degeneracy-lifted Eigenmodes with Minimum Δneff >10−4 under Different Δ

Tables Icon

Table 3. Optimal Design Supporting the Most Number of Fully Degeneracy-lifted Eigenmodes with Minimum Δneff >10−4 under Different Ellipticity when Fixing Δ = 2.5%

 figure: Fig. 2

Fig. 2 Colormaps of (a) the minimum Δneff values between adjacent modes and (b) corresponding mode number as functions of d1 and ax when fixing Δ = 2.5% and e = 2. Minimum Δneff values between adjacent modes as functions of r2 and d2 with θ respectively taking (c) 60°, (d) 90°, (e) 120°and (f) 150°when fixing m = 30%.

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Remarkably, Fig. 2(a) is not smooth and there exists some sharp peaks and dips. The main reason is that part of fiber eigenmodes would transform into each other with the change of fiber geometric parameters, such as ax and d1. That is, the order of some adjacent eigenmodes would constantly change. Obviously, before the transformation is completed, there will be a point when the two adjacent eigenmodes are alike to each other, resulting in the decrease of Δneff between these two modes [28]. Afterwards, with these two modes differing from each other, the Δneff would increase reversely, thus forming a dip. Moreover, with the increase of mode number, the transformation would exist among more eigenmodes, producing more dips with the change of fiber geometric parameters. It corresponds to the fluctuations of Δneff between two adjacent modes, which eventually leads to the dips (peaks on the contrary) in Fig. 2(a). When the BT-ERC-MMF supports fewer modes, this kind of transformation would be slight. The order of most of the eigenmodes would be fixed with the change of ax and d1.

To optimize the geometry, we then fix the structure of elliptical ring-core with Δ = 2.5%, e = 2, d1 = 4.2 µm, ax = 1.4 µm, and sweep the other four parameters. Figures 2(c)-2(f) show the colormaps of the minimum Δneff versus r2 and d2 with θ taking 60°, 90°, 120° and 150° when fixing m = 30%, respectively. As the mode number only varies within 52~54, to obtain larger minimum Δneff with reasonable parameter values, we finally choose the point m = 30%, θ = 120°, d2 = 1 µm and r2 = 50 µm as the target fiber structure size. Similarly, as shown in Fig. 2(e), within the region of d2 ≤ 2 µm and 40 µm ≤ r2 ≤ 60 µm, the fiber keeps supporting 53 eigenmodes with minimum Δneff larger than 1 × 10−4, indicating quite high tolerance.

Additional, in order to ensure accurate calculations, we verify the convergence of the simulated minimum Δneff between adjacent modes under different mesh size in the fiber elliptical ring-core region. Figure 3 shows the minimum Δneff values of the targeted BT-ERC-MMF structure at 1550 nm with mesh size varying from 1000 nm to 100 nm. When the mesh size in the fiber elliptical ring-core region is small enough, the minimum Δneff values change slightly. Hence, the practically employed mesh size of 100 nm is believed to be sufficient.

 figure: Fig. 3

Fig. 3 Minimum Δneff values of the targeted BT-ERC-MMF structure at 1550 nm with mesh size varying from 1000 nm to 100 nm.

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4. Mode properties and broadband characteristics

Specifically, we then provide the detailed stress-optical coupling effects and eigenmodes properties for the target fiber structure design. Figures 4(a) and 4(b) show the von Mises stress distribution and geometric and stress-induced birefringence (Nx-Ny) in transverse cross section of the designed BT-ERC-MMF [26]. We can see that the maximum birefringence between the outer ellipse and two SAPs is around 2.1 × 10−3. Owing to the presence of elliptical ring-core and two SAPs with low refractive indices, the intensity profiles deviate from the ring shape in conventional circular core fiber or ring-core fiber and present LP-like field distributions. In particular, the intensity profiles and electric field polarization directions (blue arrow surface) for nine eigenmodes (EM1, EM7, EM13, EM20, EM26, EM32, EM38, EM45 and EM53) are shown in Fig. 4(c). We can see that the polarization directions are all approximatively horizontal or vertical. To sufficiently illustrate the linearly polarized mode fields, the power extinction ratio (ER) of all 53 eigenmodes through a polarizer [29], which is rotated by 360 degrees, is calculated as summarized in Table 4. The ER values for most of the eigenmodes keep larger than 30 dB, except for six eigenmodes sitting in (15.58, 19.38) dB and other five eigenmodes within (26.05, 29.77) dB. Besides, Table 4 also depicts the calculated Δneff (between EMn and EMn + 1, n = 1~52), Aeff, nonlinearity and D for all fiber eigenmodes at 1550 nm. With all Δneff between adjacent modes larger than 1.59 × 10−4, the supported 53 eigenmodes are fully lifted in the designed BT-ERC-MMF. Besides, for elliptical ring-core multi-mode fiber without bow-tie SAPs, the minimum Δneff is only 6.67 × 10−6. Note that in this work, the eigenmodes in BT-ERC-MMF are denoted by EMn, where n in the subscript refers to the mode order, thus to distinguish from the fiber eigenmodes in conventional fibers.

 figure: Fig. 4

Fig. 4 (a) Von Mises stress distribution and (b) geometric and stress-induced birefringence (Nx-Ny) in transverse cross section of the designed BT-ERC-MMF. (c) The intensity profiles and electric field polarization directions (blue arrow surface) for nine eigenmodes (EM1, EM7, EM13, EM20, EM26, EM32, EM38, EM45 and EM53).

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Tables Icon

Table 4. Calculated Δneff, Aeff, γ, D and ER for all 53 Eigenmodes at 1550 nm

The effect of bending on the designed fiber is also investigated by using equivalent straight fiber models for a curved fiber based on conformal mapping [30, 31]. The minimum Δneff between any two adjacent modes and the corresponding maximum bend-induced losses (α) among all 53 eigenmodes under different fiber bend radius are calculated, along either x- or y-axis, as shown in Table 5. Note that the higher-order modes would leak out from the ring-core into the cladding with the decrease of fiber bend radius. As a result, the number of supported eigenmodes reduces to 44, 48 and 52 or 45, 48 and 51 under fiber bend radii of 1 cm, 2 cm and 5 cm along x- or y-axis, respectively. One can see that most of the maximum α are less than 9.86×10-4 dB/km along both x- and y-axes, which can be explained by the strong modal confinements resulting from the high-index ring-core. The one exception of maximum α equal to 4.75×10-3 dB/km is due to the near cut-off highest-order modes. The variations of the minimum Δneff are negligible and the minimum Δneff remains above 1.59 × 10−4 when bending along x direction, while the minimum Δneff is below 1.13×10-5 under the bend radii of 1 cm and 2 cm along y-axis. This is because that the major axis of the fiber elliptical ring-core is along the y-axis, thus a small bend radius along y direction would lead to the decrease of ellipticity which eventually reduces the modal spacing. Particularly, we find that the minimum Δneff would keep larger than 1 × 10−4 with the bend radius larger than 2.8 cm along y direction.

Tables Icon

Table 5. Mode Number, Minimum Δneff between Adjacent Modes, and Maximum Bend-induced Confinement Loss α (dB/km) along x- and y-axesa

We further investigate the wavelength dependence of the supported eigenmodes in the designed BT-ERC-MMF over the whole C band in Fig. 5. From Figs. 5(a) and 5(b), one can clearly see that all the Δneff between adjacent modes are larger than 1.09 × 10−4 over the whole C band, indicating fully degeneracy-lifted eigenmode channels for WDM transmission. Figures 5(c) and 5(d) show that the Aeff [32] for all eigenmodes slightly increases with wavelength and are within (94.8, 133.9) µm2 over the whole C band, while the fiber nonlinearity decreases with wavelength accordingly and sits in (0.7197, 1.04) km−1W−1. Note that the fiber nonlinearity is inversely proportional to the Aeff. Specially, the Aeff values of the two fundamental modes (EM1 and EM3) are around 120 µm2, which is nearly twice that in single-mode fiber of the ITU-T recommendation. Moreover, note that the mode number would decrease from 53 to 52 at wavelengths longer than 1565 nm, thus we just plot the D [33] for these 52 eigenmodes versus wavelength as shown in Fig. 5(e). In detail, the D of lower-order 50 eigenmodes over the whole C band are within (−17.8, 66.31) ps/nm/km, which is within fourfold of that in the standard single mode fiber (17 ps/nm/km). It is indicated that the chromatic dispersions can be neglected when applied in short-reach optical interconnects. On the other hand, the highest-order two eigenmodes which are operated near their cutoff wavelength, exhibit large negative D sitting in (−461.6, −113.3) ps/nm/km. Remarkably, dispersion-compensation techniques are relatively mature and can be applied for dispersion mitigation [34].

 figure: Fig. 5

Fig. 5 (a) neff, (b) Δneff, (c) Aeff, (d) nonlinearity and (e) D for all fiber eigenmodes versus wavelength over the whole C band.

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5. Discussions and conclusion

In summary, a BT-ERC-MMF supporting 53 fully degeneracy-lifted eigenmodes is designed and analyzed. The elliptical ring-core structure and two symmetrical bow-tie SAPs enables large effective refractive index separation (>10−4) over the whole C band with small chromatic dispersions, thus the BT-ERC-MMF can be compatible with existing WDM technique. This kind of fiber design would be employed in low-crosstalk direct fiber eigenmode-division multiplexing transmission with simplified or even MIMO-free processing.

Remarkably, we can also adopt PANDA-type SAPs. Generally, the fiber attenuation of bow-tie PMFs are slightly higher than the PANDA ones [35, 36]. However, the bow-tie stress-induced birefringence single-mode PMFs exhibit higher birefringence in the range of 5 × 10−4, while the PANDA-type PMFs possess lower birefringence around 3 × 10−4 [35]. For example, with certain structure size and doping concentrations, we can achieve 46 degeneracy-lifted eigenmodes with minimum Δneff between all adjacent modes larger than 1.11 × 10−4 at 1550 nm. In comparison, we eventually employ bow-tie structure to obtain larger birefringence while supporting more degeneracy-lifted eigenmodes.

The practical fabrication of the designed BT-ERC-MMF based on current fiber manufacture technologies is expected to be achievable. On one hand, the elliptical core, ring-core and elliptical ring-core fiber have been successfully drawn in the laboratory and possesses favorable data transmission performance [5,17–20]. On the other hand, the SAPs are widely used in traditional bow-tie or PANDA-type polarization-maintaining single mode fibers. Based on these well-established fabrication techniques and experiences, it could be feasible to fabricate the proposed and designed fully degeneracy-lifted BT-ERC-MMF for MIMO-free direct fiber eigenmode-division multiplexing transmission.

Funding

National Basic Research Program of China (973 Program) (2014CB340004); National Natural Science Foundation of China (NSFC) (61761130082, 11574001, 11774116, 11274131); Royal Society-Newton Advanced Fellowship; National Program for Support of Top-Notch Young Professionals; Natural Science Foundation of Hubei Province of China (2018CFA048); Program for HUST Academic Frontier Youth Team.

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References

  • View by:

  1. D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).
  2. G. Li, N. Bai, N. Zhao, and C. Xia, “Space-division multiplexing: the next frontier in optical communication,” Adv. Opt. Photonics 6, 414–487 (2014).
  3. S. Randel, R. Ryf, A. Sierra, P. J. Winzer, A. H. Gnauck, C. A. Bolle, R. J. Essiambre, D. W. Peckham, A. McCurdy, and R. Lingle., “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt. Express 19(17), 16697–16707 (2011).
    [PubMed]
  4. R. Ryf, N. K. Fontaine, H. Chen, B. Guan, B. Huang, M. Esmaeelpour, A. H. Gnauck, S. Randel, S. J. B. Yoo, A. M. J. Koonen, R. Shubochkin, Y. Sun, and R. Lingle., “Mode-multiplexed transmission over conventional graded-index multimode fibers,” Opt. Express 23(1), 235–246 (2015).
    [PubMed]
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2018 (2)

J. Wang, “Metasurfaces enabling structured light manipulation: advances and perspectives,” Chin. Opt. Lett 16(5), 050006 (2018).
[PubMed]

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light Sci. Appl. 7, 17148 (2018).

2017 (7)

2016 (4)

2015 (4)

2014 (2)

C. Brunet, P. Vaity, Y. Messaddeq, S. LaRochelle, and L. A. Rusch, “Design, fabrication and validation of an OAM fiber supporting 36 states,” Opt. Express 22(21), 26117–26127 (2014).
[PubMed]

G. Li, N. Bai, N. Zhao, and C. Xia, “Space-division multiplexing: the next frontier in optical communication,” Adv. Opt. Photonics 6, 414–487 (2014).

2013 (2)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[PubMed]

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).

2012 (1)

Y. Yue, Y. Yan, N. Ahmed, J. Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photonics J. 4, 535–543 (2012).

2011 (1)

2009 (1)

2007 (1)

O. Frazão, J. M. T. Baptista, and J. L. Santos, “Recent advances in high-birefringence fiber loop mirror sensors,” Sensors (Basel) 7(11), 2970–2983 (2007).
[PubMed]

2005 (1)

R. Guan, F. Zhu, Z. Gan, D. Huang, and S. Liu, “Stress birefringence analysis of polarization maintaining optical fibers,” Opt. Fiber Technol. 11, 240–254 (2005).

1994 (1)

C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, and A. M. Vengsarkar, “Optical fiber-based dispersion compensation using higher order modes near cutoff,” J. Lightwave Technol. 12, 1746–1758 (1994).

1986 (1)

J. Noda, K. Okamoto, and Y. Sasak, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol. 4, 1071–1089 (1986).

1984 (1)

1982 (1)

1975 (1)

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).

1973 (1)

S. H. Wemple, D. A. Pinnow, T. C. Rich, R. E. Jaeger, and L. G. Van Uitert, “Binary SiO2-B2O3 glass system: refractive index behavior and energy gap considerations,” J. Appl. Phys. 44, 5432–5437 (1973).

Ahmed, N.

Y. Yue, Y. Yan, N. Ahmed, J. Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photonics J. 4, 535–543 (2012).

Ai, J.

Bai, N.

G. Li, N. Bai, N. Zhao, and C. Xia, “Space-division multiplexing: the next frontier in optical communication,” Adv. Opt. Photonics 6, 414–487 (2014).

Baptista, J. M. T.

O. Frazão, J. M. T. Baptista, and J. L. Santos, “Recent advances in high-birefringence fiber loop mirror sensors,” Sensors (Basel) 7(11), 2970–2983 (2007).
[PubMed]

Birnbaum, K. M.

Y. Yue, Y. Yan, N. Ahmed, J. Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photonics J. 4, 535–543 (2012).

Bolle, C. A.

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[PubMed]

Brunet, C.

Cai, X.

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light Sci. Appl. 7, 17148 (2018).

Chen, H.

Chen, S.

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light Sci. Appl. 7, 17148 (2018).

L. Zhu, A. Wang, S. Chen, J. Liu, Q. Mo, C. Du, and J. Wang, “Orbital angular momentum mode groups multiplexing transmission over 2.6-km conventional multi-mode fiber,” Opt. Express 25(21), 25637–25645 (2017).
[PubMed]

A. Wang, L. Zhu, S. Chen, C. Du, Q. Mo, and J. Wang, “Characterization of LDPC-coded orbital angular momentum modes transmission and multiplexing over a 50-km fiber,” Opt. Express 24(11), 11716–11726 (2016).
[PubMed]

S. Chen, J. Liu, Y. Zhao, L. Zhu, A. Wang, S. Li, J. Du, C. Du, Q. Mo, and J. Wang, “Full-duplex bidirectional data transmission link using twisted lights multiplexing over 1.1-km orbital angular momentum fiber,” Sci. Rep. 6, 38181 (2016).
[PubMed]

Corsi, A.

Cvijetic, N.

DiGiovanni, D. J.

C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, and A. M. Vengsarkar, “Optical fiber-based dispersion compensation using higher order modes near cutoff,” J. Lightwave Technol. 12, 1746–1758 (1994).

Dolinar, S.

Y. Yue, Y. Yan, N. Ahmed, J. Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photonics J. 4, 535–543 (2012).

Dong, Y.

H. Xiao, H. Li, G. Ren, Y. Dong, S. Xiao, J. Liu, B. Wei, and S. Jian, “Polarization-Maintaining Supermode Fiber Supporting 20 Modes,” IEEE Photonics Technol. Lett. 29, 1340–1343 (2017).

Du, C.

Du, J.

S. Chen, J. Liu, Y. Zhao, L. Zhu, A. Wang, S. Li, J. Du, C. Du, Q. Mo, and J. Wang, “Full-duplex bidirectional data transmission link using twisted lights multiplexing over 1.1-km orbital angular momentum fiber,” Sci. Rep. 6, 38181 (2016).
[PubMed]

Erkmen, B. I.

Y. Yue, Y. Yan, N. Ahmed, J. Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photonics J. 4, 535–543 (2012).

Esmaeelpour, M.

Essiambre, R. J.

Feng, Z.

Fini, J. M.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).

Fleming, J. W.

Fontaine, N. K.

Frazão, O.

O. Frazão, J. M. T. Baptista, and J. L. Santos, “Recent advances in high-birefringence fiber loop mirror sensors,” Sensors (Basel) 7(11), 2970–2983 (2007).
[PubMed]

Fu, S.

Gan, Z.

R. Guan, F. Zhu, Z. Gan, D. Huang, and S. Liu, “Stress birefringence analysis of polarization maintaining optical fibers,” Opt. Fiber Technol. 11, 240–254 (2005).

Gnauck, A. H.

Guan, B.

Guan, R.

R. Guan, F. Zhu, Z. Gan, D. Huang, and S. Liu, “Stress birefringence analysis of polarization maintaining optical fibers,” Opt. Fiber Technol. 11, 240–254 (2005).

Harris, J. H.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).

Heiblum, M.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).

Huang, B.

Huang, D.

R. Guan, F. Zhu, Z. Gan, D. Huang, and S. Liu, “Stress birefringence analysis of polarization maintaining optical fibers,” Opt. Fiber Technol. 11, 240–254 (2005).

Huang, H.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[PubMed]

Y. Yue, Y. Yan, N. Ahmed, J. Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photonics J. 4, 535–543 (2012).

Ip, E.

Jaeger, R. E.

S. H. Wemple, D. A. Pinnow, T. C. Rich, R. E. Jaeger, and L. G. Van Uitert, “Binary SiO2-B2O3 glass system: refractive index behavior and energy gap considerations,” J. Appl. Phys. 44, 5432–5437 (1973).

Jian, S.

H. Xiao, H. Li, G. Ren, Y. Dong, S. Xiao, J. Liu, B. Wei, and S. Jian, “Polarization-Maintaining Supermode Fiber Supporting 20 Modes,” IEEE Photonics Technol. Lett. 29, 1340–1343 (2017).

Kanonakis, K.

Klitis, C.

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light Sci. Appl. 7, 17148 (2018).

Koonen, A. M. J.

Kristensen, P.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[PubMed]

S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett. 34(16), 2525–2527 (2009).
[PubMed]

LaRochelle, S.

Li, G.

G. Li, N. Bai, N. Zhao, and C. Xia, “Space-division multiplexing: the next frontier in optical communication,” Adv. Opt. Photonics 6, 414–487 (2014).

Li, H.

H. Xiao, H. Li, G. Ren, Y. Dong, S. Xiao, J. Liu, B. Wei, and S. Jian, “Polarization-Maintaining Supermode Fiber Supporting 20 Modes,” IEEE Photonics Technol. Lett. 29, 1340–1343 (2017).

Li, M. J.

Li, S.

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light Sci. Appl. 7, 17148 (2018).

H. Yan, S. Li, Z. Xie, X. Zheng, H. Zhang, and B. Zhou, “Design of PANDA ring-core fiber with 10 polarization-maintaining modes,” Photon. Res. 5, 1–5 (2017).

S. Chen, J. Liu, Y. Zhao, L. Zhu, A. Wang, S. Li, J. Du, C. Du, Q. Mo, and J. Wang, “Full-duplex bidirectional data transmission link using twisted lights multiplexing over 1.1-km orbital angular momentum fiber,” Sci. Rep. 6, 38181 (2016).
[PubMed]

Liang, J.

Liao, R.

Lin, J.

Liñares, J.

Lingle, R.

Liu, D.

Liu, J.

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light Sci. Appl. 7, 17148 (2018).

H. Xiao, H. Li, G. Ren, Y. Dong, S. Xiao, J. Liu, B. Wei, and S. Jian, “Polarization-Maintaining Supermode Fiber Supporting 20 Modes,” IEEE Photonics Technol. Lett. 29, 1340–1343 (2017).

L. Zhu, A. Wang, S. Chen, J. Liu, Q. Mo, C. Du, and J. Wang, “Orbital angular momentum mode groups multiplexing transmission over 2.6-km conventional multi-mode fiber,” Opt. Express 25(21), 25637–25645 (2017).
[PubMed]

S. Chen, J. Liu, Y. Zhao, L. Zhu, A. Wang, S. Li, J. Du, C. Du, Q. Mo, and J. Wang, “Full-duplex bidirectional data transmission link using twisted lights multiplexing over 1.1-km orbital angular momentum fiber,” Sci. Rep. 6, 38181 (2016).
[PubMed]

A. Wang, L. Zhu, J. Liu, C. Du, Q. Mo, and J. Wang, “Demonstration of hybrid orbital angular momentum multiplexing and time-division multiplexing passive optical network,” Opt. Express 23(23), 29457–29466 (2015).
[PubMed]

Liu, S.

R. Guan, F. Zhu, Z. Gan, D. Huang, and S. Liu, “Stress birefringence analysis of polarization maintaining optical fibers,” Opt. Fiber Technol. 11, 240–254 (2005).

Marcuse, D.

McCurdy, A.

Messaddeq, Y.

Milione, G.

Mo, Q.

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light Sci. Appl. 7, 17148 (2018).

L. Wang, J. Ai, L. Zhu, A. Wang, S. Fu, C. Du, Q. Mo, and J. Wang, “MDM transmission of CAP-16 signals over 1.1-km antipbending trench-assisted elliptical-core few-mode fiber in passive optical networks,” Opt. Express 25, 22991–23002 (2017).
[PubMed]

L. Zhu, A. Wang, S. Chen, J. Liu, Q. Mo, C. Du, and J. Wang, “Orbital angular momentum mode groups multiplexing transmission over 2.6-km conventional multi-mode fiber,” Opt. Express 25(21), 25637–25645 (2017).
[PubMed]

A. Wang, L. Zhu, S. Chen, C. Du, Q. Mo, and J. Wang, “Characterization of LDPC-coded orbital angular momentum modes transmission and multiplexing over a 50-km fiber,” Opt. Express 24(11), 11716–11726 (2016).
[PubMed]

S. Chen, J. Liu, Y. Zhao, L. Zhu, A. Wang, S. Li, J. Du, C. Du, Q. Mo, and J. Wang, “Full-duplex bidirectional data transmission link using twisted lights multiplexing over 1.1-km orbital angular momentum fiber,” Sci. Rep. 6, 38181 (2016).
[PubMed]

J. Liang, Q. Mo, S. Fu, M. Tang, P. Shum, and D. Liu, “Design and fabrication of elliptical-core few-mode fiber for MIMO-less data transmission,” Opt. Lett. 41(13), 3058–3061 (2016).
[PubMed]

A. Wang, L. Zhu, J. Liu, C. Du, Q. Mo, and J. Wang, “Demonstration of hybrid orbital angular momentum multiplexing and time-division multiplexing passive optical network,” Opt. Express 23(23), 29457–29466 (2015).
[PubMed]

Montero, C.

Moreno, V.

Nejad, R. M.

Nelson, L. E.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).

Noda, J.

J. Noda, K. Okamoto, and Y. Sasak, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol. 4, 1071–1089 (1986).

Oh, K.

Okamoto, K.

J. Noda, K. Okamoto, and Y. Sasak, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol. 4, 1071–1089 (1986).

Peckham, D. W.

Peng, G.

Pinnow, D. A.

S. H. Wemple, D. A. Pinnow, T. C. Rich, R. E. Jaeger, and L. G. Van Uitert, “Binary SiO2-B2O3 glass system: refractive index behavior and energy gap considerations,” J. Appl. Phys. 44, 5432–5437 (1973).

Poole, C. D.

C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, and A. M. Vengsarkar, “Optical fiber-based dispersion compensation using higher order modes near cutoff,” J. Lightwave Technol. 12, 1746–1758 (1994).

Prieto, X.

Ramachandran, S.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[PubMed]

S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett. 34(16), 2525–2527 (2009).
[PubMed]

Randel, S.

Ren, G.

H. Xiao, H. Li, G. Ren, Y. Dong, S. Xiao, J. Liu, B. Wei, and S. Jian, “Polarization-Maintaining Supermode Fiber Supporting 20 Modes,” IEEE Photonics Technol. Lett. 29, 1340–1343 (2017).

Ren, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[PubMed]

Y. Yue, Y. Yan, N. Ahmed, J. Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photonics J. 4, 535–543 (2012).

Rich, T. C.

S. H. Wemple, D. A. Pinnow, T. C. Rich, R. E. Jaeger, and L. G. Van Uitert, “Binary SiO2-B2O3 glass system: refractive index behavior and energy gap considerations,” J. Appl. Phys. 44, 5432–5437 (1973).

Richardson, D. J.

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N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
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Zhang, H.

Zhang, L.

Y. Yue, Y. Yan, N. Ahmed, J. Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photonics J. 4, 535–543 (2012).

Zhao, C.

Zhao, J.

Zhao, N.

G. Li, N. Bai, N. Zhao, and C. Xia, “Space-division multiplexing: the next frontier in optical communication,” Adv. Opt. Photonics 6, 414–487 (2014).

Zhao, Y.

S. Chen, J. Liu, Y. Zhao, L. Zhu, A. Wang, S. Li, J. Du, C. Du, Q. Mo, and J. Wang, “Full-duplex bidirectional data transmission link using twisted lights multiplexing over 1.1-km orbital angular momentum fiber,” Sci. Rep. 6, 38181 (2016).
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Adv. Opt. Photonics (1)

G. Li, N. Bai, N. Zhao, and C. Xia, “Space-division multiplexing: the next frontier in optical communication,” Adv. Opt. Photonics 6, 414–487 (2014).

Appl. Opt. (2)

Chin. Opt. Lett (1)

J. Wang, “Metasurfaces enabling structured light manipulation: advances and perspectives,” Chin. Opt. Lett 16(5), 050006 (2018).
[PubMed]

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IEEE Photonics Technol. Lett. (1)

H. Xiao, H. Li, G. Ren, Y. Dong, S. Xiao, J. Liu, B. Wei, and S. Jian, “Polarization-Maintaining Supermode Fiber Supporting 20 Modes,” IEEE Photonics Technol. Lett. 29, 1340–1343 (2017).

J. Appl. Phys. (1)

S. H. Wemple, D. A. Pinnow, T. C. Rich, R. E. Jaeger, and L. G. Van Uitert, “Binary SiO2-B2O3 glass system: refractive index behavior and energy gap considerations,” J. Appl. Phys. 44, 5432–5437 (1973).

J. Lightwave Technol. (2)

C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, and A. M. Vengsarkar, “Optical fiber-based dispersion compensation using higher order modes near cutoff,” J. Lightwave Technol. 12, 1746–1758 (1994).

J. Noda, K. Okamoto, and Y. Sasak, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol. 4, 1071–1089 (1986).

Light Sci. Appl. (1)

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light Sci. Appl. 7, 17148 (2018).

Nat. Photonics (1)

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).

Opt. Express (9)

L. Wang, J. Ai, L. Zhu, A. Wang, S. Fu, C. Du, Q. Mo, and J. Wang, “MDM transmission of CAP-16 signals over 1.1-km antipbending trench-assisted elliptical-core few-mode fiber in passive optical networks,” Opt. Express 25, 22991–23002 (2017).
[PubMed]

L. Wang, R. M. Nejad, A. Corsi, J. Lin, Y. Messaddeq, L. Rusch, and S. LaRochelle, “Linearly polarized vector modes: Enabling MIMO-free mode-division multiplexing,” Opt. Express 25(10), 11736–11749 (2017).
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A. Wang, L. Zhu, J. Liu, C. Du, Q. Mo, and J. Wang, “Demonstration of hybrid orbital angular momentum multiplexing and time-division multiplexing passive optical network,” Opt. Express 23(23), 29457–29466 (2015).
[PubMed]

A. Wang, L. Zhu, S. Chen, C. Du, Q. Mo, and J. Wang, “Characterization of LDPC-coded orbital angular momentum modes transmission and multiplexing over a 50-km fiber,” Opt. Express 24(11), 11716–11726 (2016).
[PubMed]

L. Zhu, A. Wang, S. Chen, J. Liu, Q. Mo, C. Du, and J. Wang, “Orbital angular momentum mode groups multiplexing transmission over 2.6-km conventional multi-mode fiber,” Opt. Express 25(21), 25637–25645 (2017).
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R. Guan, F. Zhu, Z. Gan, D. Huang, and S. Liu, “Stress birefringence analysis of polarization maintaining optical fibers,” Opt. Fiber Technol. 11, 240–254 (2005).

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Sci. Rep. (1)

S. Chen, J. Liu, Y. Zhao, L. Zhu, A. Wang, S. Li, J. Du, C. Du, Q. Mo, and J. Wang, “Full-duplex bidirectional data transmission link using twisted lights multiplexing over 1.1-km orbital angular momentum fiber,” Sci. Rep. 6, 38181 (2016).
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Science (1)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[PubMed]

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Figures (5)

Fig. 1
Fig. 1 (a) Schematic cross-section and (b) refractive index profile of the designed BT-ERC-MMF.
Fig. 2
Fig. 2 Colormaps of (a) the minimum Δneff values between adjacent modes and (b) corresponding mode number as functions of d1 and ax when fixing Δ = 2.5% and e = 2. Minimum Δneff values between adjacent modes as functions of r2 and d2 with θ respectively taking (c) 60°, (d) 90°, (e) 120°and (f) 150°when fixing m = 30%.
Fig. 3
Fig. 3 Minimum Δneff values of the targeted BT-ERC-MMF structure at 1550 nm with mesh size varying from 1000 nm to 100 nm.
Fig. 4
Fig. 4 (a) Von Mises stress distribution and (b) geometric and stress-induced birefringence (Nx-Ny) in transverse cross section of the designed BT-ERC-MMF. (c) The intensity profiles and electric field polarization directions (blue arrow surface) for nine eigenmodes (EM1, EM7, EM13, EM20, EM26, EM32, EM38, EM45 and EM53).
Fig. 5
Fig. 5 (a) neff, (b) Δneff, (c) Aeff, (d) nonlinearity and (e) D for all fiber eigenmodes versus wavelength over the whole C band.

Tables (5)

Tables Icon

Table 1 Parameters Swept to Determine the BT-ERC-MMF Structure

Tables Icon

Table 2 Optimal Design Supporting the Most Number of Fully Degeneracy-lifted Eigenmodes with Minimum Δneff >10−4 under Different Δ

Tables Icon

Table 3 Optimal Design Supporting the Most Number of Fully Degeneracy-lifted Eigenmodes with Minimum Δneff >10−4 under Different Ellipticity when Fixing Δ = 2.5%

Tables Icon

Table 4 Calculated Δneff, Aeff, γ, D and ER for all 53 Eigenmodes at 1550 nm

Tables Icon

Table 5 Mode Number, Minimum Δneff between Adjacent Modes, and Maximum Bend-induced Confinement Loss α (dB/km) along x- and y-axesa

Equations (1)

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α = ( 1 m ) α 0 + m α 1

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