Abstract

In this paper, super-modes inside multi-core fibers with circularly distributed cores are analyzed in detail. Cores are arranged within one ring, two rings, and multiple rings. Also, MCFs with a center core embedded inside the rings are discussed. In these analyses, analytical formulas are derived for the propagation constants as well as the modal distribution vectors of the super-modes.

©2014 Optical Society of America

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References

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  1. H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000).
    [Crossref] [PubMed]
  2. T. Hayashi, T. Taru, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Ultra-low-crosstalk multi-core fiber feasible to ultra-long-haul transmission,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2011, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPC2.
    [Crossref]
  3. C. Xia, N. Bai, R. Amezcua-Correa, E. Antonio-Lopez, A. Schulzgen, M. Richardson, X. Zhou, and G. Li, “Supermodes in strongly-coupled multi-core fibers,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2013, OSA Technical Digest (online) (Optical Society of America, 2013), paper OTh3K.5.
    [Crossref]
  4. C. Xia, N. Bai, I. Ozdur, X. Zhou, and G. Li, “Supermodes for optical transmission,” Opt. Express 19(17), 16653–16664 (2011).
    [Crossref] [PubMed]
  5. Y. Zheng, J. Yao, L. Zhang, Y. Wang, W. Wen, R. Zhou, Z. Di, and L. Jing, “Supermode analysis in multi-core photonic crystal fiber laser,” Proc. SPIE 7843, 784316 (2010).
  6. E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 9(4), 125–127 (1984).
    [Crossref] [PubMed]
  7. T. Mansuryan, Ph. Rigaud, G. Bouwmans, V. Kermene, Y. Quiquempois, A. Desfarges-Berthelemot, P. Armand, J. Benoist, and A. Barthélémy, “Spatially dispersive scheme for transmission and synthesis of femtosecond pulses through a multicore fiber,” Opt. Express 20(22), 24769–24777 (2012).
    [Crossref] [PubMed]
  8. J. Hudgings, L. Molter, and M. Dutta, “Design and modeling of passive optical switches and power dividers using non-planar coupled fiber arrays,” IEEE J. Quantum Electron. 36(12), 1438–1444 (2000).
    [Crossref]
  9. K. L. Reichenbach and C. Xu, “Numerical analysis of light propagation in image fibers or coherent fiber bundles,” Opt. Express 15(5), 2151–2165 (2007).
    [Crossref] [PubMed]
  10. A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, “Light evolution in arbitrary two-dimensional waveguide arrays,” Phys. Rev. A 75(5), 053814 (2007).
    [Crossref]
  11. Y. C. Meng, Q. Z. Guo, W. H. Tan, and Z. M. Huang, “Analytical solutions of coupled-mode equations for multiwaveguide systems, obtained by use of Chebyshev and generalized Chebyshev polynomials,” J. Opt. Soc. Am. A 21(8), 1518–1528 (2004).
    [Crossref] [PubMed]
  12. C. Alexeyev, T. Fadeyeva, N. Boklag, and M. Yavorsky, “Supermodes of a double-ring fibre array with symmetric coupling,” Ukrainian J. Phys. Opt. 12(2), 83–88 (2011).
    [Crossref]
  13. N. Kishi and E. Yamashita, “A simple coupled-mode analysis method for multiple-core optical fiber and coupled dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. 36(12), 1861–1868 (1988).
    [Crossref]
  14. S. Peleš, J. L. Rogers, and K. Wiesenfeld, “Robust synchronization in fiber laser arrays,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(2), 026212 (2006).
    [Crossref] [PubMed]
  15. 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]
  16. K. Takenaga, Y. Arakawa, Y. Sasaki, S. Tanigawa, S. Matsuo, K. Saitoh, and M. Koshiba, “A large effective area multi-core fiber with an optimized cladding thickness,” Opt. Express 19(26), B543–B550 (2011).
    [Crossref] [PubMed]
  17. 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]
  18. S. Matsuo, Y. Sasaki, I. Ishida, K. Takenaga, K. Saitoh, and M. Koshiba, “Multicore fiber with one-ring structure,” Proc. SPIE 8647, 86470F (2013).
  19. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62(11), 1267–1277 (1972).
    [Crossref] [PubMed]
  20. H. W. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran 77: The Art of Scientific Computing (Cambridge University, 1992), 179 pp.
  21. T. McMillen, “On the eigenvalues of double band matrices,” Linear Algebra Appl. 431(10), 1890–1897 (2009).
    [Crossref]
  22. A. Mafi and J. Moloney, “Shaping modes in multicore photonic crystal fibers,” IEEE Photonics Technol. Lett. 17(2), 348–350 (2005).
    [Crossref]

2013 (1)

S. Matsuo, Y. Sasaki, I. Ishida, K. Takenaga, K. Saitoh, and M. Koshiba, “Multicore fiber with one-ring structure,” Proc. SPIE 8647, 86470F (2013).

2012 (1)

2011 (3)

2010 (2)

Y. Zheng, J. Yao, L. Zhang, Y. Wang, W. Wen, R. Zhou, Z. Di, and L. Jing, “Supermode analysis in multi-core photonic crystal fiber laser,” Proc. SPIE 7843, 784316 (2010).

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]

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]

T. McMillen, “On the eigenvalues of double band matrices,” Linear Algebra Appl. 431(10), 1890–1897 (2009).
[Crossref]

2007 (2)

K. L. Reichenbach and C. Xu, “Numerical analysis of light propagation in image fibers or coherent fiber bundles,” Opt. Express 15(5), 2151–2165 (2007).
[Crossref] [PubMed]

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, “Light evolution in arbitrary two-dimensional waveguide arrays,” Phys. Rev. A 75(5), 053814 (2007).
[Crossref]

2006 (1)

S. Peleš, J. L. Rogers, and K. Wiesenfeld, “Robust synchronization in fiber laser arrays,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(2), 026212 (2006).
[Crossref] [PubMed]

2005 (1)

A. Mafi and J. Moloney, “Shaping modes in multicore photonic crystal fibers,” IEEE Photonics Technol. Lett. 17(2), 348–350 (2005).
[Crossref]

2004 (1)

2000 (2)

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000).
[Crossref] [PubMed]

J. Hudgings, L. Molter, and M. Dutta, “Design and modeling of passive optical switches and power dividers using non-planar coupled fiber arrays,” IEEE J. Quantum Electron. 36(12), 1438–1444 (2000).
[Crossref]

1988 (1)

N. Kishi and E. Yamashita, “A simple coupled-mode analysis method for multiple-core optical fiber and coupled dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. 36(12), 1861–1868 (1988).
[Crossref]

1984 (1)

1972 (1)

Alexeyev, C.

C. Alexeyev, T. Fadeyeva, N. Boklag, and M. Yavorsky, “Supermodes of a double-ring fibre array with symmetric coupling,” Ukrainian J. Phys. Opt. 12(2), 83–88 (2011).
[Crossref]

Arakawa, Y.

Armand, P.

Bai, N.

Barthélémy, A.

Benoist, J.

Boklag, N.

C. Alexeyev, T. Fadeyeva, N. Boklag, and M. Yavorsky, “Supermodes of a double-ring fibre array with symmetric coupling,” Ukrainian J. Phys. Opt. 12(2), 83–88 (2011).
[Crossref]

Bouwmans, G.

Desfarges-Berthelemot, A.

Di, Z.

Y. Zheng, J. Yao, L. Zhang, Y. Wang, W. Wen, R. Zhou, Z. Di, and L. Jing, “Supermode analysis in multi-core photonic crystal fiber laser,” Proc. SPIE 7843, 784316 (2010).

Dreisow, F.

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, “Light evolution in arbitrary two-dimensional waveguide arrays,” Phys. Rev. A 75(5), 053814 (2007).
[Crossref]

Dutta, M.

J. Hudgings, L. Molter, and M. Dutta, “Design and modeling of passive optical switches and power dividers using non-planar coupled fiber arrays,” IEEE J. Quantum Electron. 36(12), 1438–1444 (2000).
[Crossref]

Fadeyeva, T.

C. Alexeyev, T. Fadeyeva, N. Boklag, and M. Yavorsky, “Supermodes of a double-ring fibre array with symmetric coupling,” Ukrainian J. Phys. Opt. 12(2), 83–88 (2011).
[Crossref]

Guo, Q. Z.

Huang, Z. M.

Hudgings, J.

J. Hudgings, L. Molter, and M. Dutta, “Design and modeling of passive optical switches and power dividers using non-planar coupled fiber arrays,” IEEE J. Quantum Electron. 36(12), 1438–1444 (2000).
[Crossref]

Ishida, I.

S. Matsuo, Y. Sasaki, I. Ishida, K. Takenaga, K. Saitoh, and M. Koshiba, “Multicore fiber with one-ring structure,” Proc. SPIE 8647, 86470F (2013).

Jing, L.

Y. Zheng, J. Yao, L. Zhang, Y. Wang, W. Wen, R. Zhou, Z. Di, and L. Jing, “Supermode analysis in multi-core photonic crystal fiber laser,” Proc. SPIE 7843, 784316 (2010).

Kapon, E.

Katz, J.

Kermene, V.

Kishi, N.

N. Kishi and E. Yamashita, “A simple coupled-mode analysis method for multiple-core optical fiber and coupled dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. 36(12), 1861–1868 (1988).
[Crossref]

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]

Koshiba, M.

S. Matsuo, Y. Sasaki, I. Ishida, K. Takenaga, K. Saitoh, and M. Koshiba, “Multicore fiber with one-ring structure,” Proc. SPIE 8647, 86470F (2013).

K. Takenaga, Y. Arakawa, Y. Sasaki, S. Tanigawa, S. Matsuo, K. Saitoh, and M. Koshiba, “A large effective area multi-core fiber with an optimized cladding thickness,” Opt. Express 19(26), B543–B550 (2011).
[Crossref] [PubMed]

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]

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]

Lederer, F.

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, “Light evolution in arbitrary two-dimensional waveguide arrays,” Phys. Rev. A 75(5), 053814 (2007).
[Crossref]

Li, G.

Mafi, A.

A. Mafi and J. Moloney, “Shaping modes in multicore photonic crystal fibers,” IEEE Photonics Technol. Lett. 17(2), 348–350 (2005).
[Crossref]

Mansuryan, T.

Matsuo, S.

S. Matsuo, Y. Sasaki, I. Ishida, K. Takenaga, K. Saitoh, and M. Koshiba, “Multicore fiber with one-ring structure,” Proc. SPIE 8647, 86470F (2013).

K. Takenaga, Y. Arakawa, Y. Sasaki, S. Tanigawa, S. Matsuo, K. Saitoh, and M. Koshiba, “A large effective area multi-core fiber with an optimized cladding thickness,” Opt. Express 19(26), B543–B550 (2011).
[Crossref] [PubMed]

McMillen, T.

T. McMillen, “On the eigenvalues of double band matrices,” Linear Algebra Appl. 431(10), 1890–1897 (2009).
[Crossref]

Meng, Y. C.

Moloney, J.

A. Mafi and J. Moloney, “Shaping modes in multicore photonic crystal fibers,” IEEE Photonics Technol. Lett. 17(2), 348–350 (2005).
[Crossref]

Molter, L.

J. Hudgings, L. Molter, and M. Dutta, “Design and modeling of passive optical switches and power dividers using non-planar coupled fiber arrays,” IEEE J. Quantum Electron. 36(12), 1438–1444 (2000).
[Crossref]

Nolte, S.

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, “Light evolution in arbitrary two-dimensional waveguide arrays,” Phys. Rev. A 75(5), 053814 (2007).
[Crossref]

Ozdur, I.

Peleš, S.

S. Peleš, J. L. Rogers, and K. Wiesenfeld, “Robust synchronization in fiber laser arrays,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(2), 026212 (2006).
[Crossref] [PubMed]

Pertsch, T.

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, “Light evolution in arbitrary two-dimensional waveguide arrays,” Phys. Rev. A 75(5), 053814 (2007).
[Crossref]

Peschel, U.

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, “Light evolution in arbitrary two-dimensional waveguide arrays,” Phys. Rev. A 75(5), 053814 (2007).
[Crossref]

Quiquempois, Y.

Reichenbach, K. L.

Rigaud, Ph.

Rogers, J. L.

S. Peleš, J. L. Rogers, and K. Wiesenfeld, “Robust synchronization in fiber laser arrays,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(2), 026212 (2006).
[Crossref] [PubMed]

Saitoh, F.

Saitoh, K.

Sasaki, Y.

S. Matsuo, Y. Sasaki, I. Ishida, K. Takenaga, K. Saitoh, and M. Koshiba, “Multicore fiber with one-ring structure,” Proc. SPIE 8647, 86470F (2013).

K. Takenaga, Y. Arakawa, Y. Sasaki, S. Tanigawa, S. Matsuo, K. Saitoh, and M. Koshiba, “A large effective area multi-core fiber with an optimized cladding thickness,” Opt. Express 19(26), B543–B550 (2011).
[Crossref] [PubMed]

Snyder, W.

Stuart, H. R.

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000).
[Crossref] [PubMed]

Szameit, A.

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, “Light evolution in arbitrary two-dimensional waveguide arrays,” Phys. Rev. A 75(5), 053814 (2007).
[Crossref]

Takenaga, K.

S. Matsuo, Y. Sasaki, I. Ishida, K. Takenaga, K. Saitoh, and M. Koshiba, “Multicore fiber with one-ring structure,” Proc. SPIE 8647, 86470F (2013).

K. Takenaga, Y. Arakawa, Y. Sasaki, S. Tanigawa, S. Matsuo, K. Saitoh, and M. Koshiba, “A large effective area multi-core fiber with an optimized cladding thickness,” Opt. Express 19(26), B543–B550 (2011).
[Crossref] [PubMed]

Tan, W. H.

Tanigawa, S.

Tunnermann, A.

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, “Light evolution in arbitrary two-dimensional waveguide arrays,” Phys. Rev. A 75(5), 053814 (2007).
[Crossref]

Wang, Y.

Y. Zheng, J. Yao, L. Zhang, Y. Wang, W. Wen, R. Zhou, Z. Di, and L. Jing, “Supermode analysis in multi-core photonic crystal fiber laser,” Proc. SPIE 7843, 784316 (2010).

Wen, W.

Y. Zheng, J. Yao, L. Zhang, Y. Wang, W. Wen, R. Zhou, Z. Di, and L. Jing, “Supermode analysis in multi-core photonic crystal fiber laser,” Proc. SPIE 7843, 784316 (2010).

Wiesenfeld, K.

S. Peleš, J. L. Rogers, and K. Wiesenfeld, “Robust synchronization in fiber laser arrays,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(2), 026212 (2006).
[Crossref] [PubMed]

Xia, C.

Xu, C.

Yamashita, E.

N. Kishi and E. Yamashita, “A simple coupled-mode analysis method for multiple-core optical fiber and coupled dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. 36(12), 1861–1868 (1988).
[Crossref]

Yao, J.

Y. Zheng, J. Yao, L. Zhang, Y. Wang, W. Wen, R. Zhou, Z. Di, and L. Jing, “Supermode analysis in multi-core photonic crystal fiber laser,” Proc. SPIE 7843, 784316 (2010).

Yariv, A.

Yavorsky, M.

C. Alexeyev, T. Fadeyeva, N. Boklag, and M. Yavorsky, “Supermodes of a double-ring fibre array with symmetric coupling,” Ukrainian J. Phys. Opt. 12(2), 83–88 (2011).
[Crossref]

Zhang, L.

Y. Zheng, J. Yao, L. Zhang, Y. Wang, W. Wen, R. Zhou, Z. Di, and L. Jing, “Supermode analysis in multi-core photonic crystal fiber laser,” Proc. SPIE 7843, 784316 (2010).

Zheng, Y.

Y. Zheng, J. Yao, L. Zhang, Y. Wang, W. Wen, R. Zhou, Z. Di, and L. Jing, “Supermode analysis in multi-core photonic crystal fiber laser,” Proc. SPIE 7843, 784316 (2010).

Zhou, R.

Y. Zheng, J. Yao, L. Zhang, Y. Wang, W. Wen, R. Zhou, Z. Di, and L. Jing, “Supermode analysis in multi-core photonic crystal fiber laser,” Proc. SPIE 7843, 784316 (2010).

Zhou, X.

IEEE J. Quantum Electron. (1)

J. Hudgings, L. Molter, and M. Dutta, “Design and modeling of passive optical switches and power dividers using non-planar coupled fiber arrays,” IEEE J. Quantum Electron. 36(12), 1438–1444 (2000).
[Crossref]

IEEE Photonics Technol. Lett. (1)

A. Mafi and J. Moloney, “Shaping modes in multicore photonic crystal fibers,” IEEE Photonics Technol. Lett. 17(2), 348–350 (2005).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

N. Kishi and E. Yamashita, “A simple coupled-mode analysis method for multiple-core optical fiber and coupled dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. 36(12), 1861–1868 (1988).
[Crossref]

IEICE Electron. Express (1)

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. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Linear Algebra Appl. (1)

T. McMillen, “On the eigenvalues of double band matrices,” Linear Algebra Appl. 431(10), 1890–1897 (2009).
[Crossref]

Opt. Express (5)

Opt. Lett. (1)

Phys. Rev. A (1)

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, “Light evolution in arbitrary two-dimensional waveguide arrays,” Phys. Rev. A 75(5), 053814 (2007).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

S. Peleš, J. L. Rogers, and K. Wiesenfeld, “Robust synchronization in fiber laser arrays,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(2), 026212 (2006).
[Crossref] [PubMed]

Proc. SPIE (2)

Y. Zheng, J. Yao, L. Zhang, Y. Wang, W. Wen, R. Zhou, Z. Di, and L. Jing, “Supermode analysis in multi-core photonic crystal fiber laser,” Proc. SPIE 7843, 784316 (2010).

S. Matsuo, Y. Sasaki, I. Ishida, K. Takenaga, K. Saitoh, and M. Koshiba, “Multicore fiber with one-ring structure,” Proc. SPIE 8647, 86470F (2013).

Science (1)

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000).
[Crossref] [PubMed]

Ukrainian J. Phys. Opt. (1)

C. Alexeyev, T. Fadeyeva, N. Boklag, and M. Yavorsky, “Supermodes of a double-ring fibre array with symmetric coupling,” Ukrainian J. Phys. Opt. 12(2), 83–88 (2011).
[Crossref]

Other (3)

T. Hayashi, T. Taru, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Ultra-low-crosstalk multi-core fiber feasible to ultra-long-haul transmission,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2011, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPC2.
[Crossref]

C. Xia, N. Bai, R. Amezcua-Correa, E. Antonio-Lopez, A. Schulzgen, M. Richardson, X. Zhou, and G. Li, “Supermodes in strongly-coupled multi-core fibers,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2013, OSA Technical Digest (online) (Optical Society of America, 2013), paper OTh3K.5.
[Crossref]

H. W. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran 77: The Art of Scientific Computing (Cambridge University, 1992), 179 pp.

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Figures (9)

Fig. 1
Fig. 1 The arrangement of the fiber array and the coupling between the cores.
Fig. 2
Fig. 2 The 6-core-MCF core array arrangement.
Fig. 3
Fig. 3 The supermodes of the 6-core-MCF obtained by BPM. (a-f) mode profiles of the supermodes 1-6.
Fig. 4
Fig. 4 The supermodes of the 6-core-MCF obtained by the analytical formulas (a-f) mode profiles of the supermodes 1-6.
Fig. 5
Fig. 5 The 11-core-MCF core array arrangement and the assumed coupling coefficients between the cores.
Fig. 6
Fig. 6 The effective indexes of the super-modes of the six-core MCF. (gap size = 0.5μm).
Fig. 7
Fig. 7 The effective indexes of the super-modes of the six-core MCF. (gap size = 0μm).
Fig. 8
Fig. 8 The effective indexes of the super-modes of the six-core MCF VS Gap size.
Fig. 9
Fig. 9 The effective indexes of the super-modes of the six-core MCF VS Normalized coupling coefficient (the coupling coefficient over the free space wave number).

Tables (4)

Tables Icon

Table 1 Effective Indexes of the Supermodes Obtained by Both Methods

Tables Icon

Table 2 Eigen Values of theTwo Coupling Matrix of the MCFs with and without a Center Core

Tables Icon

Table 3 Effective Indexes of the Super-modes of the Four-core MCF (gap size = 0.5μm)

Tables Icon

Table 4 Effective Indexes of the Super-modes of the Four-core MCF (gap size = 0μm)

Equations (39)

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

dA dz =jκA A=( a 1 a N ) κ=( 0 κ κ κ 0 κ κ 0 κ κ κ 0 )
A( L )=Qexp( jDL ) Q H A( 0 )
β+2κcos( 2π( n1 ) N )
Q mn = 1 N exp( j( m1 )( n1 ) 2π N )
when N is an even number Q mn ={ 1 N n=1 2 N cos( ( m1 )( n1 ) 2π N ) 1<n N 2 1 N ( 1 ) m1 n=1+ N 2 2 N sin( ( m1 )( Nn+1 ) 2π N ) N 2 +1<nN when N is an odd number Q mn ={ 1 N n=1 2 N cos( ( m1 )( n1 ) 2π N ) 1<n N+1 2 2 N sin( ( m1 )( Nn+1 ) 2π N ) N+1 2 <nN
dA dz =jMA M=( κ 1 C C κ 2 )
κ 1 =Q D 1 Q H κ 2 =Q D 2 Q H
M= Q Total ( D 1 C C D 2 ) Q Total H Q Total =( Q 0 0 Q )
( D 1 C C D 2 )=UN U H
N=( d 1 1 c c d 2 1 d 1 N c c d 2 N )
N=VΛ V H
λ 2 ( d 1 n + d 2 n )λ+ d 1 n d 2 n c 2 =0
λ= ( d 1 n + d 2 n )± ( d 1 n d 2 n ) 2 +4 c 2 2
β+( κ 1 + κ 2 )cos( 2π( n1 ) N )± ( κ 1 κ 2 ) 2 cos ( 2π( n1 ) N ) 2 + c 2
c 2 2 k 2 +2 c 2 ±2k k 2 + c 2 ( k± k 2 + c 2 c ,1 ) k=( κ 1 κ 2 )cos( 2π( n1 ) N )
( v n 11 q n v n 12 q n v n 21 q n v n 22 q n )
β+( κ 1 + κ 2 )cos( 2π( n1 ) N )±( κ 1 κ 2 )cos( 2π( n1 ) N )
β+( 2 κ 1 cos( 2π( n1 ) N ) )± κ 1
( ± 1 2 q T , 1 2 q T ) T
M=( κ 1 C 12 0 C 21 κ 2 C 23 0 C 32 κ 3 )
M=( Q Q Q )( D 1 C 12 0 C 21 D 2 C 23 0 C 32 D 3 )( Q H Q H Q H )
( D 1 C 12 0 C 21 D 2 C 23 0 C 32 D 3 )=UN U H
N=( d 1 1 c 12 c 21 d 2 1 c 23 c 32 d 3 1 d 1 N c 12 c 21 d 2 N c 23 c 32 d 3 N )
( d 1 n c 12 c 21 d 2 n c 23 c 32 d 3 n )
λ 3 +A λ 2 +Bλ+C=0 A=( d 1 n + d 2 n + d 3 n ) B=( d 1 n d 2 n + d 1 n d 3 n + d 2 n d 3 n c 12 2 c 23 2 ) C= d 1 n d 2 n d 3 n + c 23 2 ( d 1 n + d 3 n )
λ k = 1 3 ( b+ u k D+ Δ 0 u k D )
u 1 =1, u 2 = 1+j 3 2 , u 3 = 1j 3 2 D= Δ 1 + Δ 1 2 4 Δ 0 3 2 3 Δ 0 = B 2 3c Δ 1 =2 B 3 9BC+27d
M=( κ 1 C 12 C 21 κ 2 C 23 C L,L1 κ L )
M=UN U H
( d 1 n c 12 c 21 d 2 n c 23 c L,L1 d L n )
( 2κcos( 2π( n1 ) N ) κ a κ a 2κcos( 2π( n1 ) N ) κ a κ a 2κcos( 2π( n1 ) N ) )
2 L+1 ( sin( πm L+1 ),sin( π2m L+1 ),sin( πlm L+1 ) ) T
M=( 0 b T b κ 1 C 12 C 21 κ 2 C 23 C L1,L κ L )
M 1 =( 0 b T b Q D 1 Q H )
κ 1 ± κ 1 2 +N b 0 2
( κ 1 ± κ 1 2 +N b 0 2 b 0 ,1,1,,1 ) T
M=( Q' Q Q Q )( D 1 C ' H C' D 2 C 23 C L1,L2 D L1 C L1,L C L,L1 D L )( Q ' H Q H Q H Q H )
C'= Q H ( 0 c 1,2 0 c 1,2 )Q'=( c ' + c ' 0 c 1,2 ) c ' ± = c 1,2 N N+ ( κ 1 ± κ 1 2 +N b 0 2 b 0 ) 2
( d 1 1 c ' d 1+ 1 c ' + c ' c ' + d 2 1 c L1,L2 d L1 1 c L1,L c L,L1 d L 1 )

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