Abstract

In this paper, a non-orthogonal coupled mode theory is proposed to analyze the super-modes of multi-core fibers (MCFs). The theory is valid in the strong coupling regime and can provide accurate analytical formulas for the super-modes inside MCFs. MCFs with circularly distributed cores are analyzed as an example. Analytical formulas are derived both for the refractive indexes and the eigen vectors of the super-modes. It is rigorously revealed that the eigen vectors for the super-modes of such MCFs are the row vectors of the inverse discrete Fourier transform (IDFT) matrix. Therefore, by pre-coding the signal channels via IDFT, one is able to generate the super-modes for the MCFs with circularly distributed cores.

© 2014 Optical Society of America

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2014 (1)

2013 (1)

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

2012 (1)

2011 (2)

2010 (2)

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

2009 (1)

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

2007 (1)

2006 (1)

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

2005 (2)

R. M. Gray, “Toeplitz and circulant matrices: a review,” Found. Trends Commun. Inf. Theory 2(3), 155–239 (2005).
[CrossRef]

A. Mafi, J. Moloney, “Shaping modes in multicore photonic crystal fibers,” IEEE Photon. 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, 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]

1995 (1)

B. E. Little, W. P. Huang, “Coupled-mode theory for optical waveguides,” Prog. Electromagn. Res. 10, 217–270 (1995).

1994 (1)

1984 (1)

1972 (1)

Arakawa, Y.

Armand, P.

Bai, N.

Barthélémy, A.

Benoist, J.

Bouwmans, G.

Desfarges-Berthelemot, A.

Di, Z.

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

Dutta, M.

J. Hudgings, L. Molter, 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]

Gallion, P.

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

Gray, R. M.

R. M. Gray, “Toeplitz and circulant matrices: a review,” Found. Trends Commun. Inf. Theory 2(3), 155–239 (2005).
[CrossRef]

Guo, Q. Z.

Huang, W. P.

B. E. Little, W. P. Huang, “Coupled-mode theory for optical waveguides,” Prog. Electromagn. Res. 10, 217–270 (1995).

W. P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11(3), 963–983 (1994).
[CrossRef]

Huang, Z. M.

Hudgings, J.

J. Hudgings, L. Molter, 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]

Jing, L.

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

Kapon, E.

Katz, J.

Kermene, V.

Kokubun, Y.

Y. Kokubun, 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.

Li, G.

Little, B. E.

B. E. Little, W. P. Huang, “Coupled-mode theory for optical waveguides,” Prog. Electromagn. Res. 10, 217–270 (1995).

Mafi, A.

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

Mansuryan, T.

Matsuo, S.

Meng, Y. C.

Moloney, J.

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

Molter, L.

J. Hudgings, L. Molter, 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]

Ozdur, I.

Peleš, S.

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

Quiquempois, Y.

Reichenbach, K. L.

Rigaud, Ph.

Rogers, J. L.

S. Peleš, J. L. Rogers, 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.

Snyder, W.

Stuart, H. R.

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

Takenaga, K.

Tan, W. H.

Tanigawa, S.

Wang, Y.

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

Wiesenfeld, K.

S. Peleš, J. L. Rogers, 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.

Yao, J.

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

Yariv, A.

Zhang, L.

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

Zhou, J.

J. Zhou, “Analytical formulation of super-modes inside multi-core fibers with circularly distributed cores,” Opt. Express 22(1), 673–688 (2014).
[CrossRef] [PubMed]

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

Zhou, R.

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

Zhou, X.

Found. Trends Commun. Inf. Theory (1)

R. M. Gray, “Toeplitz and circulant matrices: a review,” Found. Trends Commun. Inf. Theory 2(3), 155–239 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Hudgings, L. Molter, 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 Photon. Technol. Lett. (2)

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

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

IEICE Electron. Express (1)

Y. Kokubun, 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 (2)

Opt. Express (6)

Opt. Lett. (1)

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

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

Proc. SPIE (1)

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

Prog. Electromagn. Res. (1)

B. E. Little, W. P. Huang, “Coupled-mode theory for optical waveguides,” Prog. Electromagn. Res. 10, 217–270 (1995).

Science (1)

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

Other (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]

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]

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Figures (7)

Fig. 1
Fig. 1

Cross section of the six core fiber.

Fig. 2
Fig. 2

Mode fields calculated by BPM simulations.

Fig. 3
Fig. 3

Mode fields calculated by Eq. (17).

Fig. 4
Fig. 4

IDFT based super-mode multiplexing technique.

Fig. 5
Fig. 5

The fields of the MCF with the input vector as [1 0 0 0 0 0] before pre-coding (a) at the input of the MCF (b) after propagating inside the MCF for 1m (c) after propagating inside the MCF for 5m (d) after propagating inside the MCF for 10m.

Fig. 6
Fig. 6

IDFT based super-mode de-multiplexing technique.

Fig. 7
Fig. 7

The evolution of the fundamental super-modes of the MCF along the taper (a) at the input of the taper (b) after propagating in the taper for 100μm (c) after in the taper for 2500μm (d) after propagating in the taper for 10000μm.

Tables (2)

Tables Icon

Table 1 The calculated refractive indexes of the super-modes by the BPM, Eq. (14) and the conventional CMT

Tables Icon

Table 2 The calculated refractive indexes of the super-modes of a four core MCF by the BPM, Eq. (14) and the conventional CMT

Equations (23)

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

2 φ+ k 2 φ=0
φ= n=1 N c n φ n e jβz
2 φ n +( k n 2 β 2 ) φ n =0
n=1 N c n z φ n = n=1 N c n j( k 2 k n 2 ) 2β φ n
S c z =jEc
E mn = + + ( k 2 k n 2 ) 2β φ m φ n dxdy S mn = + + φ m φ n dxdy
E mn = E m n S mn = S m n | mn |=| m n |
E mn = E | mn |+1 S mn = S | mn |+1
E=QG Q H S=QD Q H
Q mn = 1 N exp( j 2π( m1 )( n1 ) N )
G nn = m=1 N E m exp( j 2π( n1 )( m1 ) N ) D nn = m=1 N S m exp( j 2π( n1 )( m1 ) N )
c z =jMc
M=Q D 1 G Q H
β n =β+ G nn D nn
E m = E Nm S m = S Nm
β n = β Nn+2
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
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
n eff1 = β 1 k 0 = n eff0 + k 11 + k 12 X( k 11 + k 12 ) 1 X 2 n eff2 = β 2 k 0 = n eff0 + k 11 k 12 +X( k 11 k 12 ) 1 X 2
k 11 = E 1 k 0 S 1 k 12 = E 2 k 0 S 1 X= S 2 S 1 n eff0 = β k 0
Qg
Q P Q H Q g = Q P g
P = d i a g ( exp ( j β 1 L ) , exp ( j β 2 L ) , exp ( j β N L ) )

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