Abstract

In future high capacity multicore optical fiber (MCF) networks, signal-processing devices should be able to manipulate data without sacrificing network capacity or MCFs advantages. Thus, it is crucial to have high performance novel devices that can be connected directly to MCFs without conversion to conventional single-core fibers. In this work, a novel Y-splitter for multicore optical fibers is proposed and numerically demonstrated for the first time. The splitter can directly split the power of input MCF cores by 50/50 splitting-ratio into two output MCFs cores. The splitter principle of operation mainly depends on novel double-hump graded-index (DHGI) profile that can space-division split (SDS) optical power by half. Both finite-difference-time-domain and eigenmode-expansion simulations are performed to design, verify, and characterize performance of Y-splitter. It shows wideband operation over the S, C, L, U-bands with polarization insensitivity. It also demonstrates high performance with reasonable insertion-loss, in addition to very low excess-loss and return-loss. Moreover, the splitter shows good performance tolerance to both MCF and design parameters variations.

© 2015 Optical Society of America

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References

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  1. R. Essiambre, G. Kramer, P. Winzer, G. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
    [Crossref]
  2. B. Zhu, J. Fini, M. Yan, X. Liu, S. Chandrasekhar, T. Taunay, M. Fishteyn, E. Monberg, and F. Dimarcello, “High-capacity space-division-multiplexed DWDM transmissions using multicore fiber,” J. Lightwave Technol. 30(4), 486–492 (2012).
    [Crossref]
  3. D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
    [Crossref]
  4. A. Chralyvy, “Plenary paper: The coming capacity crunch,” in European Conference on Optical Communication, 2009, Vienna, Austria, paper 1.0.2.
  5. R. Essiambre, R. Ryf, N. K. Fontaine, and S. Randel, “Breakthroughs in photonics 2012: Space-division multiplexing in multimode and multicore fibers for high-capacity optical communication,” IEEE Photonics J. 5(2), 0701307 (2013).
    [Crossref]
  6. P. Krummrich, “Spatial multiplexing for high capacity transport,” Opt. Fiber Technol. 17(5), 480–489 (2011).
    [Crossref]
  7. J. Sakaguchi, Y. Awaji, N. Wada, A. Kanno, T. Kawanishi, T. Hayashi, T. Taru, T. Kobayashi, and M. Watanabe, “109-Tb/s (7× 97× 172-Gb/s SDM/WDM/PDM) QPSK transmission through 16.8-km homogeneous multi-core fiber,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2011, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPB6.
  8. J. Sakaguchi, B. J. Puttnam, W. Klaus, Y. Awaji, N. Wada, A. Kanno, T. Kawanishi, K. Imamura, H. Inaba, K. Mukasa, R. Sugizaki, T. Kobayashi, and M. Watanabe, “305 Tb/s space division multiplexed transmission using homogeneous 19-core fiber,” J. Lightwave Technol. 31(4), 554–562 (2013).
    [Crossref]
  9. B. Zhu, T. F. Taunay, M. F. Yan, J. M. Fini, M. Fishteyn, E. M. Monberg, and F. V. Dimarcello, “Seven-core multicore fiber transmissions for passive optical network,” Opt. Express 18(11), 11117–11122 (2010).
    [Crossref] [PubMed]
  10. S. G. Leon-Saval, A. Argyros, and J. Bland-Hawthorn, “Photonic lanterns: a study of light propagation in multimode to single-mode converters,” Opt. Express 18(8), 8430–8439 (2010).
    [Crossref] [PubMed]
  11. S. G. Leon-Saval, T. A. Birks, J. Bland-Hawthorn, and M. Englund, “Multimode fiber devices with single-mode performance,” Opt. Lett. 30(19), 2545–2547 (2005).
    [Crossref] [PubMed]
  12. J. Sakaguchi, Y. Awaji, N. Wada, T. Hayashi, T. Nagashima, T. Kobayashi, and M. Watanabe, “Propagation characteristics of seven-core fiber for spatial and wavelength division multiplexed 10-Gbit/s channels,” in Optical Fiber Communication Conference /National Fiber Optic Engineers Conference 2011, OSA Technical Digest (CD) (Optical Society of America, 2011), paper OWJ2.
    [Crossref]
  13. W. Klaus, J. Sakaguchi, B. J. Puttnam, Y. Awaji, N. Wada, T. Kobayashi, and M. Watanabe, “Free-space coupling optics for multicore fibers,” IEEE Photonic Tech. L. 24(21), 1902–1905 (2012).
    [Crossref]
  14. J. Zhou and P. Gallion, “A novel mode multiplexer/demultiplexer for multi-core fibers,” IEEE Photon. Technol. Lett. 25(13), 1214–1217 (2013).
    [Crossref]
  15. Y. Kokubun and M. Koshiba, “Novel multi-core fibers for mode division multiplexing: Proposal and design principle,” IEICE Electron. Express 6(8), 522–528 (2009).
    [Crossref]
  16. F. Saitoh, K. Saitoh, and M. Koshiba, “A design method of a fiber-based mode multi/demultiplexer for mode-division multiplexing,” Opt. Express 18(5), 4709–4716 (2010).
    [Crossref] [PubMed]
  17. D. Choudhury, J. Macdonald, and K. Kar, “Ultrafast laser inscription: perspectives on future integrated applications,” Laser Photonics Rev. 8(6), 827–846 (2014).
    [Crossref]
  18. I. Kaminow, T. Li, and A. Willner, Optical Fiber Telecommunications: Systems and Networks VIB, 6th ed. (Academic Press Elsevier, 2013) , Chap. 13.
  19. M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: Proposal and design principle,” IEICE Electron. Express 6(2), 98–103 (2009).
    [Crossref]
  20. B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley Interscience, 2007), Chap. 1, 9, 24.
  21. Lumerical solutions, www.lumerical.com
  22. H. Takahashi, “Planar lightwave circuit devices for optical communication: present and future,” Proc. SPIE 5246, 520–531 (2003).
    [Crossref]
  23. M. Kawachi, “Silica waveguides on silicon and their application to integrated-optic components,” Opt. Quantum Electron. 22(5), 391–416 (1990).
    [Crossref]
  24. H. Yanagawa, T. Shimizu, S. Nakamura, and I. Ohyama, “Index-and-dimensional taper and its application to photonic devices,” J. Lightwave Technol. 10(5), 587–592 (1992).
    [Crossref]

2014 (1)

D. Choudhury, J. Macdonald, and K. Kar, “Ultrafast laser inscription: perspectives on future integrated applications,” Laser Photonics Rev. 8(6), 827–846 (2014).
[Crossref]

2013 (4)

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

R. Essiambre, R. Ryf, N. K. Fontaine, and S. Randel, “Breakthroughs in photonics 2012: Space-division multiplexing in multimode and multicore fibers for high-capacity optical communication,” IEEE Photonics J. 5(2), 0701307 (2013).
[Crossref]

J. Zhou and P. Gallion, “A novel mode multiplexer/demultiplexer for multi-core fibers,” IEEE Photon. Technol. Lett. 25(13), 1214–1217 (2013).
[Crossref]

J. Sakaguchi, B. J. Puttnam, W. Klaus, Y. Awaji, N. Wada, A. Kanno, T. Kawanishi, K. Imamura, H. Inaba, K. Mukasa, R. Sugizaki, T. Kobayashi, and M. Watanabe, “305 Tb/s space division multiplexed transmission using homogeneous 19-core fiber,” J. Lightwave Technol. 31(4), 554–562 (2013).
[Crossref]

2012 (2)

W. Klaus, J. Sakaguchi, B. J. Puttnam, Y. Awaji, N. Wada, T. Kobayashi, and M. Watanabe, “Free-space coupling optics for multicore fibers,” IEEE Photonic Tech. L. 24(21), 1902–1905 (2012).
[Crossref]

B. Zhu, J. Fini, M. Yan, X. Liu, S. Chandrasekhar, T. Taunay, M. Fishteyn, E. Monberg, and F. Dimarcello, “High-capacity space-division-multiplexed DWDM transmissions using multicore fiber,” J. Lightwave Technol. 30(4), 486–492 (2012).
[Crossref]

2011 (1)

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

2010 (4)

2009 (2)

Y. Kokubun and M. Koshiba, “Novel multi-core fibers for mode division multiplexing: Proposal and design principle,” IEICE Electron. Express 6(8), 522–528 (2009).
[Crossref]

M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: Proposal and design principle,” IEICE Electron. Express 6(2), 98–103 (2009).
[Crossref]

2005 (1)

2003 (1)

H. Takahashi, “Planar lightwave circuit devices for optical communication: present and future,” Proc. SPIE 5246, 520–531 (2003).
[Crossref]

1992 (1)

H. Yanagawa, T. Shimizu, S. Nakamura, and I. Ohyama, “Index-and-dimensional taper and its application to photonic devices,” J. Lightwave Technol. 10(5), 587–592 (1992).
[Crossref]

1990 (1)

M. Kawachi, “Silica waveguides on silicon and their application to integrated-optic components,” Opt. Quantum Electron. 22(5), 391–416 (1990).
[Crossref]

Argyros, A.

Awaji, Y.

Birks, T. A.

Bland-Hawthorn, J.

Chandrasekhar, S.

Choudhury, D.

D. Choudhury, J. Macdonald, and K. Kar, “Ultrafast laser inscription: perspectives on future integrated applications,” Laser Photonics Rev. 8(6), 827–846 (2014).
[Crossref]

Dimarcello, F.

Dimarcello, F. V.

Englund, M.

Essiambre, R.

R. Essiambre, R. Ryf, N. K. Fontaine, and S. Randel, “Breakthroughs in photonics 2012: Space-division multiplexing in multimode and multicore fibers for high-capacity optical communication,” IEEE Photonics J. 5(2), 0701307 (2013).
[Crossref]

R. Essiambre, G. Kramer, P. Winzer, G. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
[Crossref]

Fini, J.

Fini, J. M.

Fishteyn, M.

Fontaine, N. K.

R. Essiambre, R. Ryf, N. K. Fontaine, and S. Randel, “Breakthroughs in photonics 2012: Space-division multiplexing in multimode and multicore fibers for high-capacity optical communication,” IEEE Photonics J. 5(2), 0701307 (2013).
[Crossref]

Foschini, G.

Gallion, P.

J. Zhou and P. Gallion, “A novel mode multiplexer/demultiplexer for multi-core fibers,” IEEE Photon. Technol. Lett. 25(13), 1214–1217 (2013).
[Crossref]

Goebel, B.

Imamura, K.

Inaba, H.

Kanno, A.

Kar, K.

D. Choudhury, J. Macdonald, and K. Kar, “Ultrafast laser inscription: perspectives on future integrated applications,” Laser Photonics Rev. 8(6), 827–846 (2014).
[Crossref]

Kawachi, M.

M. Kawachi, “Silica waveguides on silicon and their application to integrated-optic components,” Opt. Quantum Electron. 22(5), 391–416 (1990).
[Crossref]

Kawanishi, T.

Klaus, W.

Kobayashi, T.

Kokubun, Y.

Y. Kokubun and M. Koshiba, “Novel multi-core fibers for mode division multiplexing: Proposal and design principle,” IEICE Electron. Express 6(8), 522–528 (2009).
[Crossref]

M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: Proposal and design principle,” IEICE Electron. Express 6(2), 98–103 (2009).
[Crossref]

Koshiba, M.

F. Saitoh, K. Saitoh, and M. Koshiba, “A design method of a fiber-based mode multi/demultiplexer for mode-division multiplexing,” Opt. Express 18(5), 4709–4716 (2010).
[Crossref] [PubMed]

M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: Proposal and design principle,” IEICE Electron. Express 6(2), 98–103 (2009).
[Crossref]

Y. Kokubun and M. Koshiba, “Novel multi-core fibers for mode division multiplexing: Proposal and design principle,” IEICE Electron. Express 6(8), 522–528 (2009).
[Crossref]

Kramer, G.

Krummrich, P.

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

Leon-Saval, S. G.

Liu, X.

Macdonald, J.

D. Choudhury, J. Macdonald, and K. Kar, “Ultrafast laser inscription: perspectives on future integrated applications,” Laser Photonics Rev. 8(6), 827–846 (2014).
[Crossref]

Monberg, E.

Monberg, E. M.

Mukasa, K.

Nakamura, S.

H. Yanagawa, T. Shimizu, S. Nakamura, and I. Ohyama, “Index-and-dimensional taper and its application to photonic devices,” J. Lightwave Technol. 10(5), 587–592 (1992).
[Crossref]

Nelson, L.

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

Ohyama, I.

H. Yanagawa, T. Shimizu, S. Nakamura, and I. Ohyama, “Index-and-dimensional taper and its application to photonic devices,” J. Lightwave Technol. 10(5), 587–592 (1992).
[Crossref]

Puttnam, B. J.

Randel, S.

R. Essiambre, R. Ryf, N. K. Fontaine, and S. Randel, “Breakthroughs in photonics 2012: Space-division multiplexing in multimode and multicore fibers for high-capacity optical communication,” IEEE Photonics J. 5(2), 0701307 (2013).
[Crossref]

Richardson, D.

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

Ryf, R.

R. Essiambre, R. Ryf, N. K. Fontaine, and S. Randel, “Breakthroughs in photonics 2012: Space-division multiplexing in multimode and multicore fibers for high-capacity optical communication,” IEEE Photonics J. 5(2), 0701307 (2013).
[Crossref]

Saitoh, F.

Saitoh, K.

F. Saitoh, K. Saitoh, and M. Koshiba, “A design method of a fiber-based mode multi/demultiplexer for mode-division multiplexing,” Opt. Express 18(5), 4709–4716 (2010).
[Crossref] [PubMed]

M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: Proposal and design principle,” IEICE Electron. Express 6(2), 98–103 (2009).
[Crossref]

Sakaguchi, J.

Shimizu, T.

H. Yanagawa, T. Shimizu, S. Nakamura, and I. Ohyama, “Index-and-dimensional taper and its application to photonic devices,” J. Lightwave Technol. 10(5), 587–592 (1992).
[Crossref]

Sugizaki, R.

Takahashi, H.

H. Takahashi, “Planar lightwave circuit devices for optical communication: present and future,” Proc. SPIE 5246, 520–531 (2003).
[Crossref]

Taunay, T.

Taunay, T. F.

Wada, N.

Watanabe, M.

Winzer, P.

Yan, M.

Yan, M. F.

Yanagawa, H.

H. Yanagawa, T. Shimizu, S. Nakamura, and I. Ohyama, “Index-and-dimensional taper and its application to photonic devices,” J. Lightwave Technol. 10(5), 587–592 (1992).
[Crossref]

Zhou, J.

J. Zhou and P. Gallion, “A novel mode multiplexer/demultiplexer for multi-core fibers,” IEEE Photon. Technol. Lett. 25(13), 1214–1217 (2013).
[Crossref]

Zhu, B.

IEEE Photon. Technol. Lett. (1)

J. Zhou and P. Gallion, “A novel mode multiplexer/demultiplexer for multi-core fibers,” IEEE Photon. Technol. Lett. 25(13), 1214–1217 (2013).
[Crossref]

IEEE Photonic Tech. L. (1)

W. Klaus, J. Sakaguchi, B. J. Puttnam, Y. Awaji, N. Wada, T. Kobayashi, and M. Watanabe, “Free-space coupling optics for multicore fibers,” IEEE Photonic Tech. L. 24(21), 1902–1905 (2012).
[Crossref]

IEEE Photonics J. (1)

R. Essiambre, R. Ryf, N. K. Fontaine, and S. Randel, “Breakthroughs in photonics 2012: Space-division multiplexing in multimode and multicore fibers for high-capacity optical communication,” IEEE Photonics J. 5(2), 0701307 (2013).
[Crossref]

IEICE Electron. Express (2)

M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: Proposal and design principle,” IEICE Electron. Express 6(2), 98–103 (2009).
[Crossref]

Y. Kokubun and M. Koshiba, “Novel multi-core fibers for mode division multiplexing: Proposal and design principle,” IEICE Electron. Express 6(8), 522–528 (2009).
[Crossref]

J. Lightwave Technol. (4)

Laser Photonics Rev. (1)

D. Choudhury, J. Macdonald, and K. Kar, “Ultrafast laser inscription: perspectives on future integrated applications,” Laser Photonics Rev. 8(6), 827–846 (2014).
[Crossref]

Nat. Photonics (1)

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

Opt. Express (3)

Opt. Fiber Technol. (1)

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

Opt. Lett. (1)

Opt. Quantum Electron. (1)

M. Kawachi, “Silica waveguides on silicon and their application to integrated-optic components,” Opt. Quantum Electron. 22(5), 391–416 (1990).
[Crossref]

Proc. SPIE (1)

H. Takahashi, “Planar lightwave circuit devices for optical communication: present and future,” Proc. SPIE 5246, 520–531 (2003).
[Crossref]

Other (6)

J. Sakaguchi, Y. Awaji, N. Wada, T. Hayashi, T. Nagashima, T. Kobayashi, and M. Watanabe, “Propagation characteristics of seven-core fiber for spatial and wavelength division multiplexed 10-Gbit/s channels,” in Optical Fiber Communication Conference /National Fiber Optic Engineers Conference 2011, OSA Technical Digest (CD) (Optical Society of America, 2011), paper OWJ2.
[Crossref]

J. Sakaguchi, Y. Awaji, N. Wada, A. Kanno, T. Kawanishi, T. Hayashi, T. Taru, T. Kobayashi, and M. Watanabe, “109-Tb/s (7× 97× 172-Gb/s SDM/WDM/PDM) QPSK transmission through 16.8-km homogeneous multi-core fiber,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2011, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPB6.

A. Chralyvy, “Plenary paper: The coming capacity crunch,” in European Conference on Optical Communication, 2009, Vienna, Austria, paper 1.0.2.

I. Kaminow, T. Li, and A. Willner, Optical Fiber Telecommunications: Systems and Networks VIB, 6th ed. (Academic Press Elsevier, 2013) , Chap. 13.

B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley Interscience, 2007), Chap. 1, 9, 24.

Lumerical solutions, www.lumerical.com

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Figures (7)

Fig. 1
Fig. 1 The MCF Y-splitter structure. (a) The 3D schematic diagram of Y-Splitter with seven sub-splitters connected between 7-core MCFs, each sub-splitter consists of 3-stages. (b) Schematic front-view of Y-splitter showing transverse cross-section of rectangular waveguides and MCFs cores/ cladding.

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Fig. 2
Fig. 2 A single sub Y-splitter structure and its FDTD simulation. (a) 2D refractive index profile of sub Y-splitter. The inset shows the double-hump graded index profile of the second stage space-division-splitter (SDS). The white arrows indicate approximate locations of total internal reflections occurrence. (b) The FDTD simulation of electric-filed magnitude (|E| a.u.) for a single sub Y-splitter at the wavelength of 1555nm.

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Fig. 3
Fig. 3 The EME simulations (|E| a.u.) of Y-splitters together with MCFs cores. (a)-(g) show the longitudinal cross-sections of the seven sub Y-splitters connected to input/ output core 1 to 7, respectively.

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Fig. 4
Fig. 4 The EME simulations (|E| a.u.) of Y-splitters together with MCFs cores. It shows transverse cross-sections of: (a) Input MCF, (b) Lenses, (c) SDS, (d) Separators, and (e) Output MCFs. The cross-sections correspond to x-positions in Fig. 3 equal to 350, 750, 1750, 2500, and 3500 µm, respectively.

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Fig. 5
Fig. 5 The wideband performance evaluation parameters of Y-Splitter for each core. (a) Insertion-loss, (b) Excess-loss, (c) Splitting-ratio, (d) Power-imbalance, (e) Polarization-dependent-loss, and (f) Return-loss.

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Fig. 6
Fig. 6 The EL and PI performance tolerance to variations in MCF parameters. (a) Input MCF misalignment in ‘y’ and ‘z’ directions, (b) Output MCF misalignment in ‘y’ and ‘z’ directions, (c) Variations in MCF core diameters, (d) Variations in MCF core refractive indices.

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Fig. 7
Fig. 7 The performance tolerance to variations in Y-splitter design parameters. (a) The EL tolerance to variations in different stages lengths, (b) The EL and PI tolerances to misalignment between stages positions and/ or graded-indices profiles.

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Tables (1)

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Table 1 The summary of ray-trajectory theoretical analysis results

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Equations (9)

Equations on this page are rendered with MathJax. Learn more.

n 1 ( y ) = n o – 8∗ 10 −6 y 2
n 2 ( y ) = n o − 4.4∗ 10 −6 (y±20) 2
n 3 ( y ) = n o − 4.4∗ 10 −6 (y±78.5) 2
d 2 y d x 2 = 1 n( y ) dn( y ) dy
n( y )= n o ( 1− α m 2 2 y 2 )
y om ≅ y c +( y im − y c ) Cos( αx )+ θ im α  Sin( αx )
θ om = d y om dx ≅−( y im − y c ) Sin( αx )+ θ im  Cos( αx )
y i, m = y o, m−1
n m ( y i, m ) . Sin θ i, m = n m−1 ( y o, m−1 ) . Sin θ o, m−1

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