## Few-mode fiber with inverse-parabolic graded-index profile for transmission of OAM-carrying modes |

Optics Express, Vol. 22, Issue 15, pp. 18044-18055 (2014)

http://dx.doi.org/10.1364/OE.22.018044

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### Abstract

A novel type of few-mode fiber, characterized by an inverse-parabolic graded-index profile, is proposed for the robust transmission of cylindrical vector modes as well as modes carrying quantized orbital angular momentum (OAM). Large effective index separations between vector modes (>2.1 × 10^{−4}) are numerically calculated and experimentally confirmed in this fiber over the whole C-band, enabling transmission of OAM(+/−1,1) modes for distances up to 1.1 km. Simple design rules are provided for the optimization of the fiber parameters.

© 2014 Optical Society of America

## 1. Introduction

1. R.-J. Essiambre and R. W. Tkach, “Capacity Trends and Limits of Optical Communication Networks,” Proc. IEEE **100**(5), 1035–1055 (2012). [CrossRef]

2. D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics **7**(5), 354–362 (2013). [CrossRef]

3. P. M. Krummrich, “Spatial multiplexing for high capacity transport,” Opt. Fiber Technol. **17**(5), 480–489 (2011). [CrossRef]

4. P. J. Winzer, “Energy-Efficient Optical Transport Capacity Scaling Through Spatial Multiplexing,” IEEE Photon. Technol. Lett. **23**(13), 851–853 (2011). [CrossRef]

*scalar*LP modes are actually composed of several spatial and polarization degenerate modes that randomly couple during propagation, their demultiplexing at the fiber output usually requires multiple-input and multiple-output (MIMO) signal processing whose complexity scales quadratically with the number of modes. An alternate scheme for MDM is to excite fiber modes carrying quantized states of orbital angular momentum (OAM) [5]. The OAM state of an optical mode stems from its helical phase front − also known as an optical vortex − whose formulation inside a FMF relies on utilizing distinct

*vector modes*(i.e. the true eigenmodes) of the cylindrical fiber [6

6. P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of Orbital Angular Momentum Transfer between Acoustic and Optical Vortices in Optical Fiber,” Phys. Rev. Lett. **96**(4), 043604 (2006). [CrossRef] [PubMed]

*n*, between the effective indices of adjacent modes. A so-called “vortex fiber” characterized by a high-index ring was thus shown to separate the effective indices of the constitutive vector modes of the LP

_{eff}_{11}group by Δ

*n*≈1.8 × 10

_{eff}^{−4}[7

7. 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). [CrossRef] [PubMed]

8. 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). [CrossRef] [PubMed]

*n*≈1.0 × 10

_{eff}^{−4}was shown to support 12 OAM modes [9

9. P. Gregg, P. Kristensen, S. E. Golowich, J. Ø. Olsen, P. Steinvurzel, and S. Ramachandran, “Stable Transmission of 12 OAM States in Air-Core Fiber,” in Proc. of CLEO: 2012, CTu2K (2013). [CrossRef]

10. C. Brunet, B. Ung, Y. Messaddeq, S. LaRochelle, E. Bernier, and L. A. Rusch, “Design of an Optical Fiber Supporting 16 OAM Modes,” in Proc. of OFC: 2014, Th2A.24 (2014). [CrossRef]

11. 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 Photon. J. **4**(2), 535–543 (2012). [CrossRef]

12. S. Li and J. Wang, “A Compact Trench-Assisted Multi-Orbital-Angular-Momentum Multi-Ring Fiber for Ultrahigh-Density Space-Division Multiplexing (19 Rings × 22 Modes),” Sci. Rep. **4**, 3853 (2014). [CrossRef] [PubMed]

13. Y. Yue, L. Zhang, Y. Yan, N. Ahmed, J.-Y. Yang, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Octave-spanning supercontinuum generation of vortices in an As2S3 ring photonic crystal fiber,” Opt. Lett. **37**(11), 1889–1891 (2012). [CrossRef] [PubMed]

14. B. Ung, P. Vaity, L. Wang, Y. Messaddeq, L. A. Rusch, and S. LaRochelle, “Inverse-parabolic graded-index profile for transmission of cylindrical vector modes in optical fibers,” in Proc. of OFC: 2014, Tu3K.4 (2014). [CrossRef]

*n*> 2.1 × 10

_{eff}^{−4}) as predicted by theory and confirmed in experiments. The proposed FMF is characterized by an inverse-parabolic graded-refractive-index profile. We first present the

*inverse-parabolic graded-index fiber*(IPGIF) and theoretically investigate how this fiber can achieve wide modal separations. A discussion of the fabrication procedure and the characterization methods used to confirm the large effective index splittings, is then presented. We finally demonstrate the propagation of OAM carrying modes in the IPGIF over a kilometer length, thus supporting the appeal of this design for MDM with OAM states.

## 2. Description of the IPGIF

*n*>1 × 10

_{eff}^{−4}) it was demonstrated that fiber designs tailored towards large refractive index gradients coupled with high modal field gradients are imperative [7

7. 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). [CrossRef] [PubMed]

7. 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). [CrossRef] [PubMed]

12. S. Li and J. Wang, “A Compact Trench-Assisted Multi-Orbital-Angular-Momentum Multi-Ring Fiber for Ultrahigh-Density Space-Division Multiplexing (19 Rings × 22 Modes),” Sci. Rep. **4**, 3853 (2014). [CrossRef] [PubMed]

15. R. L. Lachance and P.-A. Bélanger, “Modes in Divergent Parabolic Graded-Index Optical Fibers,” J. Lightwave Technol. **9**(11), 1425–1430 (1991). [CrossRef]

*n*

_{1}and

*n*

_{2}are respectively the refractive indices at the core center (

*r*= 0) and in the cladding (

*r*>

*a*). Also,

*n*denotes the refractive index value exactly at the core-cladding interface (

_{a}*r*=

*a*). For the inverse-parabolic profiles of interest, with

*N*<0, the maximum refractive index contrast occurs at the core-cladding interface and is given by

*N*= 0 corresponds to the conventional step-index profile. Figure 1 shows the refractive index profile of the IPGIF specifically studied in this work with

*N*= −4, Δ

*n*

_{max}= 0.05 and

*a =*3 μm.

*n*

_{2}= 1.4440 at λ = 1550 nm), we performed numerical simulations based on the finite-element-method (FEM) with a commercial software (COMSOL). The IPGIF considered in Fig. 1 was found to support up to three mode groups (LP

_{01}, LP

_{11}and LP

_{21}). The field profiles of the constitutive vector modes are shown in Fig. 2.

## 3. Analysis of the effective index separation between vector modes

### 3.1 Theoretical analysis of the modal separations in the IPGIF

### 3.2 Numerical analysis of the modal separations in the IPGIF

*N*≤ 0) and (0.02 ≤ Δ

*n*

_{max}≤ 0.07). For simplification purposes, the optimization solely focuses on maximizing the intermodal index separation (Δ

*n*) between the modes of the LP

_{eff}_{11}group (TE

_{01}, HE

_{21}, and TM

_{01}). However, it is expected that higher-order modes sharing the same radial mode number,

*m*= 1 (e.g. EH

_{11}and HE

_{31}), will similarly achieve large intermodal separations.

_{11}mode group as a function of the profile curvature and maximum index contrast. The results show that Δ

*n*indeed scales with

_{eff}*n*

_{max}, thus indicating that the sharpest profiles with the highest refractive-index-contrasts are those that enable the largest effective index separation of vector modes. Moreover, the calculations in Fig. 3(a) suggest that very large effective index separations, Δ

*n*>4.0 × 10

_{eff}^{−4}, may theoretically be achieved with an IPGIF profile.

*n*

_{max}, the relationship between Δ

*n*and -

_{eff}*N*takes a sigmoid behavior that grows towards a saturation value, Δ

*n*

_{eff,}_{max}. One could argue that an optimum profile curvature is close to

*N*= −3 since more than 95% of the Δ

*n*

_{eff,}_{max}value is reached at this point.

*N*, the Δ

*n*increases quasi-linearly with the relative permittivity contrast,

_{eff}*N*= −4 and

*a*, down to an optimal core radius, that occurs just before modal cutoff. Hence the validity of the

*N*= −4 and some practical values of the maximum refractive index contrast:

*a*=

*a*

_{max}) for

*N*= −4 and Δ

*n*

_{max}= 0.05 is indeed

*a*

_{max}= 3 μm, as chosen in our target IPGIF design shown in Fig. 1. Inspection of our simulation data in Fig. 4(d) indicate that the minimum effective index separations occur between the TM

_{01}and HE

_{21}modes when

*a*<

*a*

_{max}, and between the TE

_{01}and HE

_{21}modes when

*a*>

*a*

_{max}. Crucially, it also means that all the vector modes {TE

_{01}, HE

_{21}and TM

_{01}} of the LP

_{11}group are evenly spaced apart at the optimal core radius

*a*=

*a*

_{max}.

## 4. Calculation of bend induced losses in IPGIFs

17. M. Heiblum and J. H. Harris, “Analysis of Curved Optical Waveguides by Conformal Transformation,” IEEE J. Quantum Electron. **11**(2), 75–83 (1975). [CrossRef]

18. D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. **21**(23), 4208–4213 (1982). [CrossRef] [PubMed]

19. C. Schulze, A. Lorenz, D. Flamm, A. Hartung, S. Schröter, H. Bartelt, and M. Duparré, “Mode resolved bend loss in few-mode optical fibers,” Opt. Express **21**(3), 3170–3181 (2013). [CrossRef] [PubMed]

20. J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area,” Opt. Express **14**(1), 69–81 (2006). [CrossRef] [PubMed]

*n*, to obtain the

_{fiber}*equivalent index*profile,

*n*, generated by a fiber bend,

_{eq}*R*, applied along the transverse

_{bend}*x*-axis direction:where the 1.40 factor was added in the denominator to account for the additional change in refractive index due to photoelastic effects created by the local strain in bent fused silica fibers, as derived in [19

19. C. Schulze, A. Lorenz, D. Flamm, A. Hartung, S. Schröter, H. Bartelt, and M. Duparré, “Mode resolved bend loss in few-mode optical fibers,” Opt. Express **21**(3), 3170–3181 (2013). [CrossRef] [PubMed]

*confinement losses*of the fundamental HE

_{11}mode were numerically computed for a tight bend of

*R*= 1 cm radius. The results are shown in Fig. 5 as a function of the profile curvature and for different refractive index contrasts. We also note that bend-induced confinement losses of the higher-order modes within the LP

_{bend}_{11}group are approximately an order of magnitude larger compared to the fundamental HE

_{11}mode.

*N*). Secondly, the bend-induced losses rapidly decrease as the core-cladding index contrast Δ

*n*

_{max}is raised. In particular for the high-index contrast profiles of practical interest (Δ

*n*

_{max}> 0.02 and −4 ≤

*N*≤ −1), modal confinement within the core is so strong that bend-induced confinement losses become negligible, in agreement with prior analysis [20

20. J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area,” Opt. Express **14**(1), 69–81 (2006). [CrossRef] [PubMed]

21. M. E. Lines, W. A. Reed, D. J. DiGiovanni, and J. R. Hamblin, “Explanation of anomalous loss in high delta singlemode fibres,” Electron. Lett. **35**(12), 1009–1010 (1999). [CrossRef]

## 5. Fabrication and characterization of the IPGIF

*N*= −4, Δ

*n*

_{max}= 0.05 and

*a*= 3 μm, corresponding to the refractive index profile presented in Fig. 1. These IPGIF parameters theoretically enable large intermodal index separations of

*n*

_{max}= 0.05) that is still readily attainable in germania-doped silica glass preforms manufactured by the MCVD process.

_{2}dopants in gaseous state inside a fused silica tube − so as to coat the interior walls − which was then collapsed to produce the macroscopic all-glass preform. The refractive index profile was first measured on the fiber preform. The small spatial diffusion of the germania dopants (that occur during fiber drawing at 2000 °C) was then simulated through Fick's second law of diffusion and the Sellmeier equation of GeO

_{2}-doped silica glass [22

22. J. W. Fleming, “Dispersion in GeO2-SiO2 glasses,” Appl. Opt. **23**(24), 4486–4493 (1984). [CrossRef] [PubMed]

*n*

_{max}= 0.0431. The refractive index profile was also measured directly on the fabricated fiber using a refracted near-field analyzer (Exfo NR-9200HR). Although these on-fiber measurements were in agreement with the downscaled preform profile of Fig. 6, they were not retained for subsequent modal simulations because of their limited spatial resolution, 0.1 μm, which precluded from accurately resolving the sharp refractive index features of the fabricated IPGIF.

*n*) and group velocity dispersions (GVD) of all guided modes over the C-band are plotted in Fig. 7(a)-7(b). The dispersion curves in Fig. 7(b) were calculated with the formula

_{eff}_{0}= 1550 nm.

*n*≈2.1 × 10

_{eff}^{−4}inside the LP

_{11}group (TE

_{01}, HE

_{21}, TM

_{01}) and Δ

*n*≈1.6 × 10

_{eff}^{−4}for the LP

_{21}group (EH

_{11}, HE

_{31}) throughout the C-band. GVD values [Fig. 7(b)] for the fundamental HE

_{11}and the LP

_{11}mode group vary between 4 and 12 ps/(nm-km), while those of the LP

_{21}mode group lies in the strongly normal dispersion regime (<-100 ps/(nm-km)) and exhibits a pronounced negative slope. The latter behavior may be explained by the relatively close proximity of the LP

_{21}mode group to the cutoff (i.e. the index of undoped silica cladding) as shown in Fig. 7(a), hence resulting in greater group velocity dispersion.

23. T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. **15**(8), 1277–1294 (1997). [CrossRef]

*λ*= 1575.31, 1571.34, and 1554.01 nm respectively. The narrow peaks (

_{p}*λ*

_{k}_{-}

*) stem from the cross-coupling of a given mode “*

_{p}*p*” with another counter-propagating mode “

*k*”. Values of the effective indices for each vector mode can be retrieved either through

*n*=

_{eff,p}*λ*/Λ (where

_{p}*λ*is the wavelength at the self-reflection peak) or alternatively via

_{p}*n*= (2

_{eff,k}*λ*

_{k}_{-}

*-*

_{p}*λ*)/Λ where

_{p}*λ*

_{k}_{-}

*is the cross-coupling peak of mode “*

_{p}*k*” with a lower-order mode “

*p*” having a self-reflection peak at

*λ*.

_{p}*n*≈3.9 × 10

_{eff}^{−4}inside the LP

_{11}mode group (#2), and Δ

*n*≈1.9 × 10

_{eff}^{−4}in the LP

_{21}mode group (#3). The latter extracted Δ

*n*values are somewhat larger than what was predicted by FEM calculations [Fig. 7(a)]. These discrepancies can be attributed to the limited precision in the reconstructed refractive index profile of the fiber (based on the initial preform), and the inevitable presence of small longitudinal variations in the refractive index profile of the drawn fiber. Moreover, the UV side-writing process of the FBG may raise the local refractive index which can slightly perturb the ideal modal properties.

_{eff}24. L. Wang, P. Vaity, B. Ung, Y. Messaddeq, L. A. Rusch, and S. LaRochelle, “Characterization of OAM fibers using fiber Bragg gratings,” Opt. Express **22**(13), 15653–15661 (2014). [CrossRef] [PubMed]

24. L. Wang, P. Vaity, B. Ung, Y. Messaddeq, L. A. Rusch, and S. LaRochelle, “Characterization of OAM fibers using fiber Bragg gratings,” Opt. Express **22**(13), 15653–15661 (2014). [CrossRef] [PubMed]

^{−4}inside the LP

_{11}mode group and 1.57 × 10

^{−4}for the LP

_{21}mode group. Discrepancies (<0.8 × 10

^{−4}) between the effective index splittings measured with the weaker FBG and those calculated by FEM [in Fig. 7(a)] are within experimental errors

## 6. Transmission of OAM modes

*even*and

*odd*hybrid modes:

*m*denotes the radial mode number (i.e. the number of rings in the mode profile). Although strictly speaking the fundamental

_{-1,1}, OAM

_{-2,1}and OAM

_{+2,1}modes − excited with corresponding input OAM beams of left(-) circular polarization − were captured at the fiber output by a CCD camera. We recall that for this specific circular polarization (-sign), the detected OAM

_{-2,1}and OAM

_{+2,1}modes originate from the superposition of the even and odd π/2-shifted copies of the HE

_{31}and EH

_{11}eigenmodes, respectively.

_{+1,1}mode (excited by right( + ) circular polarized light) after a propagation distance of 1.1 km in the IPGIF, as shown on the right-hand side of Fig. 10. Due to the proximity of the LP

_{21}mode group to the cutoff, the OAM

_{± 2,1}modes experienced higher losses that prevented us from reliably detecting their presence after >100 m distance. The propagation losses averaged over all mode groups, were measured by the cutback method and estimated at 8.6 dB/km. Finer measurements utilizing selective excitation of the fundamental HE

_{11}mode indicate that this mode sustains lower losses, on the order of 6.5 dB/km. We are currently investigating the origin of these losses and whether they could be reduced by an optimization of the fiber fabrication process.

## 7. Conclusion

*inverse-parabolic graded-index fiber*(IPGIF) that enables very large effective index separations between its supported vector modes. In particular, we numerically studied and experimentally demonstrated an IPGIF design enabling over Δ

*n*>2.1 × 10

_{eff}^{−4}separation between the vector modes of the LP

_{11}mode group {TE

_{01}, HE

_{21}, TM

_{01}}. Subsequently, we experimentally showed the transmission of the OAM

_{± 1,1}mode over more than 1 km. The flexibility of this design and its remarkable ability to lift modal degeneracy and separate the vector modes into distinct communication channels, make the IPGIF a promising fiber towards achieving a practical long-distance OAM based mode-division multiplexing system.

## Acknowledgments

## References and links

1. | R.-J. Essiambre and R. W. Tkach, “Capacity Trends and Limits of Optical Communication Networks,” Proc. IEEE |

2. | D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics |

3. | P. M. Krummrich, “Spatial multiplexing for high capacity transport,” Opt. Fiber Technol. |

4. | P. J. Winzer, “Energy-Efficient Optical Transport Capacity Scaling Through Spatial Multiplexing,” IEEE Photon. Technol. Lett. |

5. | A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. |

6. | P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of Orbital Angular Momentum Transfer between Acoustic and Optical Vortices in Optical Fiber,” Phys. Rev. Lett. |

7. | S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett. |

8. | 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 |

9. | P. Gregg, P. Kristensen, S. E. Golowich, J. Ø. Olsen, P. Steinvurzel, and S. Ramachandran, “Stable Transmission of 12 OAM States in Air-Core Fiber,” in Proc. of CLEO: 2012, CTu2K (2013). [CrossRef] |

10. | C. Brunet, B. Ung, Y. Messaddeq, S. LaRochelle, E. Bernier, and L. A. Rusch, “Design of an Optical Fiber Supporting 16 OAM Modes,” in Proc. of OFC: 2014, Th2A.24 (2014). [CrossRef] |

11. | 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 Photon. J. |

12. | S. Li and J. Wang, “A Compact Trench-Assisted Multi-Orbital-Angular-Momentum Multi-Ring Fiber for Ultrahigh-Density Space-Division Multiplexing (19 Rings × 22 Modes),” Sci. Rep. |

13. | Y. Yue, L. Zhang, Y. Yan, N. Ahmed, J.-Y. Yang, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Octave-spanning supercontinuum generation of vortices in an As2S3 ring photonic crystal fiber,” Opt. Lett. |

14. | B. Ung, P. Vaity, L. Wang, Y. Messaddeq, L. A. Rusch, and S. LaRochelle, “Inverse-parabolic graded-index profile for transmission of cylindrical vector modes in optical fibers,” in Proc. of OFC: 2014, Tu3K.4 (2014). [CrossRef] |

15. | R. L. Lachance and P.-A. Bélanger, “Modes in Divergent Parabolic Graded-Index Optical Fibers,” J. Lightwave Technol. |

16. | J. Bures, |

17. | M. Heiblum and J. H. Harris, “Analysis of Curved Optical Waveguides by Conformal Transformation,” IEEE J. Quantum Electron. |

18. | D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. |

19. | C. Schulze, A. Lorenz, D. Flamm, A. Hartung, S. Schröter, H. Bartelt, and M. Duparré, “Mode resolved bend loss in few-mode optical fibers,” Opt. Express |

20. | J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area,” Opt. Express |

21. | M. E. Lines, W. A. Reed, D. J. DiGiovanni, and J. R. Hamblin, “Explanation of anomalous loss in high delta singlemode fibres,” Electron. Lett. |

22. | J. W. Fleming, “Dispersion in GeO2-SiO2 glasses,” Appl. Opt. |

23. | T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. |

24. | L. Wang, P. Vaity, B. Ung, Y. Messaddeq, L. A. Rusch, and S. LaRochelle, “Characterization of OAM fibers using fiber Bragg gratings,” Opt. Express |

**OCIS Codes**

(060.2280) Fiber optics and optical communications : Fiber design and fabrication

(060.2330) Fiber optics and optical communications : Fiber optics communications

(060.4230) Fiber optics and optical communications : Multiplexing

(050.4865) Diffraction and gratings : Optical vortices

**ToC Category:**

Fiber Optics

**History**

Original Manuscript: May 23, 2014

Revised Manuscript: July 2, 2014

Manuscript Accepted: July 6, 2014

Published: July 17, 2014

**Citation**

B. Ung, P. Vaity, L. Wang, Y. Messaddeq, L. A. Rusch, and S. LaRochelle, "Few-mode fiber with inverse-parabolic graded-index profile for transmission of OAM-carrying modes," Opt. Express **22**, 18044-18055 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-15-18044

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### References

- R.-J. Essiambre and R. W. Tkach, “Capacity Trends and Limits of Optical Communication Networks,” Proc. IEEE100(5), 1035–1055 (2012). [CrossRef]
- D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics7(5), 354–362 (2013). [CrossRef]
- P. M. Krummrich, “Spatial multiplexing for high capacity transport,” Opt. Fiber Technol.17(5), 480–489 (2011). [CrossRef]
- P. J. Winzer, “Energy-Efficient Optical Transport Capacity Scaling Through Spatial Multiplexing,” IEEE Photon. Technol. Lett.23(13), 851–853 (2011). [CrossRef]
- A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon.3, 161–204 (2011).
- P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of Orbital Angular Momentum Transfer between Acoustic and Optical Vortices in Optical Fiber,” Phys. Rev. Lett.96(4), 043604 (2006). [CrossRef] [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). [CrossRef] [PubMed]
- 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,” Science340(6140), 1545–1548 (2013). [CrossRef] [PubMed]
- P. Gregg, P. Kristensen, S. E. Golowich, J. Ø. Olsen, P. Steinvurzel, and S. Ramachandran, “Stable Transmission of 12 OAM States in Air-Core Fiber,” in Proc. of CLEO: 2012, CTu2K (2013). [CrossRef]
- C. Brunet, B. Ung, Y. Messaddeq, S. LaRochelle, E. Bernier, and L. A. Rusch, “Design of an Optical Fiber Supporting 16 OAM Modes,” in Proc. of OFC: 2014, Th2A.24 (2014). [CrossRef]
- 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 Photon. J.4(2), 535–543 (2012). [CrossRef]
- S. Li and J. Wang, “A Compact Trench-Assisted Multi-Orbital-Angular-Momentum Multi-Ring Fiber for Ultrahigh-Density Space-Division Multiplexing (19 Rings × 22 Modes),” Sci. Rep.4, 3853 (2014). [CrossRef] [PubMed]
- Y. Yue, L. Zhang, Y. Yan, N. Ahmed, J.-Y. Yang, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Octave-spanning supercontinuum generation of vortices in an As2S3 ring photonic crystal fiber,” Opt. Lett.37(11), 1889–1891 (2012). [CrossRef] [PubMed]
- B. Ung, P. Vaity, L. Wang, Y. Messaddeq, L. A. Rusch, and S. LaRochelle, “Inverse-parabolic graded-index profile for transmission of cylindrical vector modes in optical fibers,” in Proc. of OFC: 2014, Tu3K.4 (2014). [CrossRef]
- R. L. Lachance and P.-A. Bélanger, “Modes in Divergent Parabolic Graded-Index Optical Fibers,” J. Lightwave Technol.9(11), 1425–1430 (1991). [CrossRef]
- J. Bures, Guided Optics: Optical Fibers and All-Fiber Components (Wiley-VCH, 2009), Chap. 5.
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