Abstract

The propagation dynamics of 7-core multi-core fibers (MCFs) with identical and three-types of cores are analytically derived based on the coupled-mode theory. The mode coupling dynamics can be aperiodic with transmission distance for MCF with identical cores. For MCFs with heterogeneous cores, it is found that even though signals from different core groups will not couple with each other, the coupling within their own group is significantly affected by the presence of other core groups. Joint signal processing techniques to mitigate mode coupling induced-cross-talks such as least mean square (LMS) algorithm and maximum likelihood (ML) detection are investigated and corresponding transmission performance are determined for coherent as well as intensity modulated formats. It is shown that aperiodic mode coupling in intensity modulated systems induces cross-talks that are difficult to eliminate through signal processing. The analytical insights may help in optimizing MCF designs and corresponding signal processing techniques for future high capacity MCF transmission systems.

© 2012 OSA

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2011 (5)

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 multicore fibres: proposal and design principle,” IEICE Electron. Express 6(2), 98–103 (2009).
[CrossRef]

2008 (1)

2007 (2)

N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, “Mode competition in multi-core fiber amplifier,” Opt. Commun. 277(2), 390–396 (2007).
[CrossRef]

A. P. T. Lau, L. Xu, and T. Wang, “Performance of receivers and detection algorithms for modal multiplexing in multimode fiber systems,” IEEE Photon. Technol. Lett. 19(14), 1087–1089 (2007).
[CrossRef]

2006 (1)

C. P. Tsekrekos, A. Martinez, F. M. Huijskens, and A. M. J. Koonen, “Design considerations for transparent mode group diversity multiplexing,” IEEE Photon. Technol. Lett. 18(22), 2359–2361 (2006).
[CrossRef]

2004 (1)

L. A. B. Windover, J. N. Simom, S. A. Rosenau, K. S. Giboney, G. M. Flower, L. W. Mirkarimi, A. Grot, B. Law, C. K. Lin, A. Tandon, R. W. Gruhlke, H. Xia, G. Rankin, M. R. T. Tan, and D. W. Dolfi, “Parallel optical interconnections > 100 Gb/s,” J. Lightwave Technol. 22, 20055–22063 (2004).

1998 (1)

P. Glas, M. Naumann, A. Schirrmacher, and Th. Pertsch, “The multicore fiber—a novel design for a diode pumped fiber laser,” Opt. Commun. 151(1-3), 187–195 (1998).
[CrossRef]

1986 (1)

N. Kishi, E. Yamashita, and K. Atsuki, “Modal and coupling-field analysis of optical fibers with circularly distributed multiple cores and a central core,” J. Lightwave Technol. 4(8), 991–996 (1986).
[CrossRef]

1983 (1)

H. A. Haus and L. Molter-Orr, “Coupled multiple waveguide systems,” IEEE J. Quantum Electron. 19(5), 840–844 (1983).
[CrossRef]

1972 (1)

Al Amin, A.

Atsuki, K.

N. Kishi, E. Yamashita, and K. Atsuki, “Modal and coupling-field analysis of optical fibers with circularly distributed multiple cores and a central core,” J. Lightwave Technol. 4(8), 991–996 (1986).
[CrossRef]

Bai, N.

Barros, D. J. F.

Bolle, C. A.

Chandrasekhar, S.

Chen, S.

Chen, X.

Dimarcello, F. V.

Dolfi, D. W.

L. A. B. Windover, J. N. Simom, S. A. Rosenau, K. S. Giboney, G. M. Flower, L. W. Mirkarimi, A. Grot, B. Law, C. K. Lin, A. Tandon, R. W. Gruhlke, H. Xia, G. Rankin, M. R. T. Tan, and D. W. Dolfi, “Parallel optical interconnections > 100 Gb/s,” J. Lightwave Technol. 22, 20055–22063 (2004).

Elkin, N. N.

N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, “Mode competition in multi-core fiber amplifier,” Opt. Commun. 277(2), 390–396 (2007).
[CrossRef]

Essiambre, R.-J.

Fini, J. M.

Fishteyn, M.

Flower, G. M.

L. A. B. Windover, J. N. Simom, S. A. Rosenau, K. S. Giboney, G. M. Flower, L. W. Mirkarimi, A. Grot, B. Law, C. K. Lin, A. Tandon, R. W. Gruhlke, H. Xia, G. Rankin, M. R. T. Tan, and D. W. Dolfi, “Parallel optical interconnections > 100 Gb/s,” J. Lightwave Technol. 22, 20055–22063 (2004).

Gao, G.

Giboney, K. S.

L. A. B. Windover, J. N. Simom, S. A. Rosenau, K. S. Giboney, G. M. Flower, L. W. Mirkarimi, A. Grot, B. Law, C. K. Lin, A. Tandon, R. W. Gruhlke, H. Xia, G. Rankin, M. R. T. Tan, and D. W. Dolfi, “Parallel optical interconnections > 100 Gb/s,” J. Lightwave Technol. 22, 20055–22063 (2004).

Glas, P.

P. Glas, M. Naumann, A. Schirrmacher, and Th. Pertsch, “The multicore fiber—a novel design for a diode pumped fiber laser,” Opt. Commun. 151(1-3), 187–195 (1998).
[CrossRef]

Gnauck, A. H.

Grot, A.

L. A. B. Windover, J. N. Simom, S. A. Rosenau, K. S. Giboney, G. M. Flower, L. W. Mirkarimi, A. Grot, B. Law, C. K. Lin, A. Tandon, R. W. Gruhlke, H. Xia, G. Rankin, M. R. T. Tan, and D. W. Dolfi, “Parallel optical interconnections > 100 Gb/s,” J. Lightwave Technol. 22, 20055–22063 (2004).

Gruhlke, R. W.

L. A. B. Windover, J. N. Simom, S. A. Rosenau, K. S. Giboney, G. M. Flower, L. W. Mirkarimi, A. Grot, B. Law, C. K. Lin, A. Tandon, R. W. Gruhlke, H. Xia, G. Rankin, M. R. T. Tan, and D. W. Dolfi, “Parallel optical interconnections > 100 Gb/s,” J. Lightwave Technol. 22, 20055–22063 (2004).

Haus, H. A.

H. A. Haus and L. Molter-Orr, “Coupled multiple waveguide systems,” IEEE J. Quantum Electron. 19(5), 840–844 (1983).
[CrossRef]

Huijskens, F. M.

C. P. Tsekrekos, A. Martinez, F. M. Huijskens, and A. M. J. Koonen, “Design considerations for transparent mode group diversity multiplexing,” IEEE Photon. Technol. Lett. 18(22), 2359–2361 (2006).
[CrossRef]

Imamura, K.

K. Imamura, K. Mukasa, and T. Yagi, “Investigation on multi-core fibers with large Aeff and low micro bending loss,” OFC 2010, OWK6 (2010).

Ip, E.

Kahn, J. M.

Kishi, N.

N. Kishi, E. Yamashita, and K. Atsuki, “Modal and coupling-field analysis of optical fibers with circularly distributed multiple cores and a central core,” J. Lightwave Technol. 4(8), 991–996 (1986).
[CrossRef]

Kokubun, Y.

M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multicore fibres: 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]

Koonen, A. M. J.

C. P. Tsekrekos, A. Martinez, F. M. Huijskens, and A. M. J. Koonen, “Design considerations for transparent mode group diversity multiplexing,” IEEE Photon. Technol. Lett. 18(22), 2359–2361 (2006).
[CrossRef]

Koshiba, M.

M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Multi-core fiber design and analysis: coupled-mode theory and coupled-power theory,” Opt. Express 19(26), B102–B111 (2011).
[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]

M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multicore fibres: proposal and design principle,” IEICE Electron. Express 6(2), 98–103 (2009).
[CrossRef]

Lau, A. P. T.

E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express 16(2), 753–791 (2008).
[CrossRef] [PubMed]

A. P. T. Lau, L. Xu, and T. Wang, “Performance of receivers and detection algorithms for modal multiplexing in multimode fiber systems,” IEEE Photon. Technol. Lett. 19(14), 1087–1089 (2007).
[CrossRef]

Law, B.

L. A. B. Windover, J. N. Simom, S. A. Rosenau, K. S. Giboney, G. M. Flower, L. W. Mirkarimi, A. Grot, B. Law, C. K. Lin, A. Tandon, R. W. Gruhlke, H. Xia, G. Rankin, M. R. T. Tan, and D. W. Dolfi, “Parallel optical interconnections > 100 Gb/s,” J. Lightwave Technol. 22, 20055–22063 (2004).

Li, A.

Li, G.

Lin, C. K.

L. A. B. Windover, J. N. Simom, S. A. Rosenau, K. S. Giboney, G. M. Flower, L. W. Mirkarimi, A. Grot, B. Law, C. K. Lin, A. Tandon, R. W. Gruhlke, H. Xia, G. Rankin, M. R. T. Tan, and D. W. Dolfi, “Parallel optical interconnections > 100 Gb/s,” J. Lightwave Technol. 22, 20055–22063 (2004).

Lingle, R.

Liu, X.

Martinez, A.

C. P. Tsekrekos, A. Martinez, F. M. Huijskens, and A. M. J. Koonen, “Design considerations for transparent mode group diversity multiplexing,” IEEE Photon. Technol. Lett. 18(22), 2359–2361 (2006).
[CrossRef]

Matsuo, S.

McCurdy, A.

Mirkarimi, L. W.

L. A. B. Windover, J. N. Simom, S. A. Rosenau, K. S. Giboney, G. M. Flower, L. W. Mirkarimi, A. Grot, B. Law, C. K. Lin, A. Tandon, R. W. Gruhlke, H. Xia, G. Rankin, M. R. T. Tan, and D. W. Dolfi, “Parallel optical interconnections > 100 Gb/s,” J. Lightwave Technol. 22, 20055–22063 (2004).

Molter-Orr, L.

H. A. Haus and L. Molter-Orr, “Coupled multiple waveguide systems,” IEEE J. Quantum Electron. 19(5), 840–844 (1983).
[CrossRef]

Monberg, E. M.

Mukasa, K.

K. Imamura, K. Mukasa, and T. Yagi, “Investigation on multi-core fibers with large Aeff and low micro bending loss,” OFC 2010, OWK6 (2010).

Napartovich, A. P.

N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, “Mode competition in multi-core fiber amplifier,” Opt. Commun. 277(2), 390–396 (2007).
[CrossRef]

Naumann, M.

P. Glas, M. Naumann, A. Schirrmacher, and Th. Pertsch, “The multicore fiber—a novel design for a diode pumped fiber laser,” Opt. Commun. 151(1-3), 187–195 (1998).
[CrossRef]

Ozdur, I.

Peckham, D. W.

Pertsch, Th.

P. Glas, M. Naumann, A. Schirrmacher, and Th. Pertsch, “The multicore fiber—a novel design for a diode pumped fiber laser,” Opt. Commun. 151(1-3), 187–195 (1998).
[CrossRef]

Randel, S.

Rankin, G.

L. A. B. Windover, J. N. Simom, S. A. Rosenau, K. S. Giboney, G. M. Flower, L. W. Mirkarimi, A. Grot, B. Law, C. K. Lin, A. Tandon, R. W. Gruhlke, H. Xia, G. Rankin, M. R. T. Tan, and D. W. Dolfi, “Parallel optical interconnections > 100 Gb/s,” J. Lightwave Technol. 22, 20055–22063 (2004).

Rosenau, S. A.

L. A. B. Windover, J. N. Simom, S. A. Rosenau, K. S. Giboney, G. M. Flower, L. W. Mirkarimi, A. Grot, B. Law, C. K. Lin, A. Tandon, R. W. Gruhlke, H. Xia, G. Rankin, M. R. T. Tan, and D. W. Dolfi, “Parallel optical interconnections > 100 Gb/s,” J. Lightwave Technol. 22, 20055–22063 (2004).

Ryf, R.

Saitoh, K.

M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Multi-core fiber design and analysis: coupled-mode theory and coupled-power theory,” Opt. Express 19(26), B102–B111 (2011).
[CrossRef] [PubMed]

M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multicore fibres: proposal and design principle,” IEICE Electron. Express 6(2), 98–103 (2009).
[CrossRef]

Schirrmacher, A.

P. Glas, M. Naumann, A. Schirrmacher, and Th. Pertsch, “The multicore fiber—a novel design for a diode pumped fiber laser,” Opt. Commun. 151(1-3), 187–195 (1998).
[CrossRef]

Shieh, W.

Sierra, A.

Simom, J. N.

L. A. B. Windover, J. N. Simom, S. A. Rosenau, K. S. Giboney, G. M. Flower, L. W. Mirkarimi, A. Grot, B. Law, C. K. Lin, A. Tandon, R. W. Gruhlke, H. Xia, G. Rankin, M. R. T. Tan, and D. W. Dolfi, “Parallel optical interconnections > 100 Gb/s,” J. Lightwave Technol. 22, 20055–22063 (2004).

Snyder, A. W.

Takenaga, K.

Tan, M. R. T.

L. A. B. Windover, J. N. Simom, S. A. Rosenau, K. S. Giboney, G. M. Flower, L. W. Mirkarimi, A. Grot, B. Law, C. K. Lin, A. Tandon, R. W. Gruhlke, H. Xia, G. Rankin, M. R. T. Tan, and D. W. Dolfi, “Parallel optical interconnections > 100 Gb/s,” J. Lightwave Technol. 22, 20055–22063 (2004).

Tandon, A.

L. A. B. Windover, J. N. Simom, S. A. Rosenau, K. S. Giboney, G. M. Flower, L. W. Mirkarimi, A. Grot, B. Law, C. K. Lin, A. Tandon, R. W. Gruhlke, H. Xia, G. Rankin, M. R. T. Tan, and D. W. Dolfi, “Parallel optical interconnections > 100 Gb/s,” J. Lightwave Technol. 22, 20055–22063 (2004).

Taunay, T. F.

Troshchieva, V. N.

N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, “Mode competition in multi-core fiber amplifier,” Opt. Commun. 277(2), 390–396 (2007).
[CrossRef]

Tsekrekos, C. P.

C. P. Tsekrekos, A. Martinez, F. M. Huijskens, and A. M. J. Koonen, “Design considerations for transparent mode group diversity multiplexing,” IEEE Photon. Technol. Lett. 18(22), 2359–2361 (2006).
[CrossRef]

Vysotsky, D. V.

N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, “Mode competition in multi-core fiber amplifier,” Opt. Commun. 277(2), 390–396 (2007).
[CrossRef]

Wang, T.

F. Yaman, N. Bai, B. Zhu, T. Wang, and G. Li, “Long distance transmission in few-mode fibers,” Opt. Express 18(12), 13250–13257 (2010).
[CrossRef] [PubMed]

A. P. T. Lau, L. Xu, and T. Wang, “Performance of receivers and detection algorithms for modal multiplexing in multimode fiber systems,” IEEE Photon. Technol. Lett. 19(14), 1087–1089 (2007).
[CrossRef]

Windover, L. A. B.

L. A. B. Windover, J. N. Simom, S. A. Rosenau, K. S. Giboney, G. M. Flower, L. W. Mirkarimi, A. Grot, B. Law, C. K. Lin, A. Tandon, R. W. Gruhlke, H. Xia, G. Rankin, M. R. T. Tan, and D. W. Dolfi, “Parallel optical interconnections > 100 Gb/s,” J. Lightwave Technol. 22, 20055–22063 (2004).

Winzer, P. J.

Xia, C.

Xia, H.

L. A. B. Windover, J. N. Simom, S. A. Rosenau, K. S. Giboney, G. M. Flower, L. W. Mirkarimi, A. Grot, B. Law, C. K. Lin, A. Tandon, R. W. Gruhlke, H. Xia, G. Rankin, M. R. T. Tan, and D. W. Dolfi, “Parallel optical interconnections > 100 Gb/s,” J. Lightwave Technol. 22, 20055–22063 (2004).

Xu, L.

A. P. T. Lau, L. Xu, and T. Wang, “Performance of receivers and detection algorithms for modal multiplexing in multimode fiber systems,” IEEE Photon. Technol. Lett. 19(14), 1087–1089 (2007).
[CrossRef]

Yagi, T.

K. Imamura, K. Mukasa, and T. Yagi, “Investigation on multi-core fibers with large Aeff and low micro bending loss,” OFC 2010, OWK6 (2010).

Yaman, F.

Yamashita, E.

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J. M. Fini, B. Zhu, T. F. Taunay, and M. F. Yan, “Bends in the design of low-crosstalk multicore fiber communications links,” OECC 2010, 7C2–3 (2010).

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K. Takenaga, Y. Arakawa, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “Reduction of crosstalk by trench-assisted multi-core fiber,” OFC 2011, OWJ4 (2011).

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Figures (10)

Fig. 1
Fig. 1

Homogeneous 7-core MCF arranged in a triangular lattice with pitch Λ and core radius a.

Fig. 2
Fig. 2

Propagation dynamics of a homogeneous 7-core MCF with Λ = 30 μm, a = 4.5 μm, and Δ = 0.370% for the case of light launching into core 2. The aperiodic coupling dynamics cannot be predicted by a 2-core model, thus indicating its inadequacy for the analysis of mode coupling and cross-talks in MCFs as MIMO transmission systems.

Fig. 3
Fig. 3

Generalized coupling lengths c of a homogeneous 7-core MCF as a function of the relative index difference Δ for pitch values of 30, 35, and 40 μm. The core radius is 4.5 μm.

Fig. 4
Fig. 4

Heterogeneous 7-core MCF with three types of cores: Δ2 = Δ4 = Δ6, Δ3 = Δ5 = Δ7, and Δ1 ≠ Δ2 ≠ Δ3.

Fig. 5
Fig. 5

Propagation dynamics for a 7-core MCF with Δ1 = 0.370%, Δ2 = Δ4 = Δ6 = 0.325%, Δ3 = Δ5 = Δ7 = 0.360%, Λ = 30 μm, and a = 4.5 μm when light is launched into core 2. The coupling dynamics for a corresponding homogeneous 3-core MCF in the absence of core 1, 3, 5, and 7 is also shown. The coupling features of the 7-core MCF resemble those of a homogeneous 3-core MCF but the generalized coupling length are very different.

Fig. 6
Fig. 6

Dependence of c2 and c3 on Δ2 for various combinations of Δ1 and Δ3. The generalized coupling lengths decrease when the cores become similar and increase when the eigenvalues of the coupling matrix R (or propagation constants of the eigenmodes of the composite MCF) given in Eq. (5) become similar.

Fig. 7
Fig. 7

Convergence behavior using the LMS algorithm for equalizing mode coupling induced cross-talks for a homogenous 7-core fiber system with QPSK signals. The SNR is 20 dB and the propagation distance is z = c.

Fig. 8
Fig. 8

Convergence behavior using the LMS algorithm for equalizing mode coupling induced cross-talks for a homogenous 7-core fiber system with OOK signals and direct detection. The SNR is 20 dB and the propagation distance is z = c.

Fig. 9
Fig. 9

BER lower bound for a homogeneous 7-core MCF with OOK modulation format and joint ML detection as a function of transmission distance. As the mode coupling dynamics is aperiodic, the lower bound is also aperiodic with z and transmission performance does not approach back to that at z = 0.

Fig. 10
Fig. 10

Schematic diagram of a heterogeneous 4-core MCF consisting of two groups of identical cores: Δ2 = Δ4 and Δ3 = Δ5.

Equations (53)

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dA( z ) dz =CA( z )
c pq ={ j C pq exp[ j( β p β q )z ] pq, 0 p=q,
C pq = 2 Δ q W q U p K 0 ( W p d pq a p ) [ a p U q J 1 ( U q ) I 0 ( W p a q a p ) + a q W p J 0 ( U q ) I 1 ( W p a q a p ) ] [ V p J 1 ( U q ) K 1 ( W p )( a p 2 U q 2 + a q 2 W p 2 ) ] 1
E p (z)= A p (z) exp(j β p z).
dE( z ) dz =RE( z )
r pq ={ j C pq pq, j β p p=q.
E(z)=V [ exp( γ p z) δ pq ] V 1 E(0)
V= [ v 1 v 2 ... v n ]
C=j[ 0 C 12 C 12 C 12 C 12 C 12 C 12 C 12 0 C 12 0 0 0 C 12 C 12 C 12 0 C 12 0 0 0 C 12 0 C 12 0 C 12 0 0 C 12 0 0 C 12 0 C 12 0 C 12 0 0 0 C 12 0 C 12 C 12 C 12 0 0 0 C 12 0 ]
γ 1 =j C 12 (1+ 7 ) ,   γ 2 = γ 3 =j C 12  ,   γ 4 = γ 5 =j C 12  ,   γ 6 =j C 12 (1 7 ) ,   γ 7 =2j C 12  
A 1 ( z )=[ cos( 7 C 12 z )+ j 7 sin( 7 C 12 z ) ]exp( j C 12 z )
A p ( z )= j 7 sin( 7 C 12 z )exp( j C 12 z ) p1
| A 1 ( z ) | 2 = 1 7 + 6 7 cos 2 ( 7 C 12 z )
| A p ( z ) | 2 = 1 7 sin 2 ( 7 C 12 z ) p1,
c1 = π 2 7 C 12 = π | j( γ 1 γ 6 ) | ,
| A 1 ( c1 ) | 2 = 1 7 .
A 1 ( z )= j 7 sin( 7 C 12 z )exp( j C 12 z ),
A 2 ( z )= 2 3 cos( C 12 z )+ 1 6 exp( 2j C 12 z )+ 1 6 exp( j C 12 z )[ cos( 7 C 12 z ) j 7 sin( 7 C 12 z ) ],
A 3 ( z )= A 7 ( z )= j 3 sin( C 12 z ) 1 6 exp( 2j C 12 z )+ 1 6 exp( j C 12 z )[ cos( 7 C 12 z ) j 7 sin( 7 C 12 z ) ],
A 4 ( z )= A 6 ( z )= 1 3 cos( C 12 z )+ 1 6 exp( 2j C 12 z )+ 1 6 exp( j C 12 z )[ cos( 7 C 12 z ) j 7 sin( 7 C 12 z ) ],
A 5 ( z )=j 2 3 sin( C 12 z ) 1 6 exp( 2j C 12 z )+ 1 6 exp( j C 12 z )[ cos( 7 C 12 z ) j 7 sin( 7 C 12 z ) ].
c2 = 1.585/ C 12
| A 2 ( c2 ) | 2 = 2.040 × 10 -2
C=j×[ 0 C 12 e jΔ β 12 z C 13 e jΔ β 13 z C 12 e jΔ β 12 z C 13 e jΔ β 13 z C 12 e jΔ β 12 z C 13 e jΔ β 13 z C 21 e jΔ β 21 z 0 C 23 e jΔ β 23 z C 24 0 C 24 C 23 e jΔ β 23 z C 31 e jΔ β 31 z C 32 e jΔ β 32 z 0 C 32 e jΔ β 32 z C 35 0 C 35 C 21 e jΔ β 21 z C 24 C 23 e jΔ β 23 z 0 C 23 e jΔ β 23 z C 24 0 C 31 e jΔ β 31 z 0 C 35 C 32 e jΔ β 32 z 0 C 32 e jΔ β 32 z C 35 C 21 e jΔ β 21 z C 24 0 C 24 C 23 e jΔ β 23 z 0 C 23 e jΔ β 23 z C 31 e jΔ β 31 z C 32 e jΔ β 32 z C 35 0 C 35 C 32 e jΔ β 32 z 0 ]
γ 1 = γ 6 =j[ α ¯ 2 + α ¯ 3 + ( α ¯ 2 α ¯ 3 ) 2 + C 23 C 32 + β 1 ], γ 2 = γ 7 =j[ α ¯ 2 + α ¯ 3 ( α ¯ 2 α ¯ 3 ) 2 + C 23 C 32 + β 1 ], γ 3 =j( g+h+ α 2 + α 3 3 + β 1 ), γ 4 = 3 2 ( gh )+j[ 1 2 ( g+h )+ α 2 + α 3 3 + β 1 ], γ 5 = 3 2 ( gh )+j[ 1 2 ( g+h )+ α 2 + α 3 3 + β 1 ]
α 2 = β 2 β 1 +2 C 24 , α 3 = β 3 β 1 +2 C 35 ,
α ¯ 2 = β 2 β 1 C 24 2 , α ¯ 3 = β 3 β 1 C 35 2 ,
u= α 2 α 3 3( C 12 C 21 + C 13 C 31 )4 C 23 C 32 1 3 ( α 2 + α 3 ) 2 ,
v= 2 27 ( α 2 + α 3 ) 3 + 1 3 ( α 2 + α 3 )[ α 2 α 3 3( C 12 C 21 + C 13 C 31 )4 C 23 C 32 ] +3( C 13 C 31 α 2 + C 12 C 21 α 3 )6( C 32 C 13 C 21 + C 12 C 23 C 31 ),
g= { v 2 + [ ( u 3 ) 3 + ( v 2 ) 2 ] 1/2 } 1/3 , h= { v 2 [ ( u 3 ) 3 + ( v 2 ) 2 ] 1/2 } 1/3 .
{ | A 2 ( z ) | 2 = 1 9 + 8 9 cos 2 [ j( γ 2 γ 5 )z 2 ], | A 4 ( z ) | 2 = | A 6 ( z ) | 2 = 4 9 sin 2 [ j( γ 2 γ 5 )z 2 ] for Δ 2 < Δ 1 , Δ 3 | A 2 ( z ) | 2 = 1 9 + 8 9 cos 2 [ j( γ 1 γ 3 )z 2 ], | A 4 ( z ) | 2 = | A 6 ( z ) | 2 = 4 9 sin 2 [ j( γ 1 γ 3 )z 2 ] for Δ 2 > Δ 1 , Δ 3 | A 2 ( z ) | 2 = 1 9 + 8 9 cos 2 [ j( γ 2 γ 4 )z 2 ], | A 4 ( z ) | 2 = | A 6 ( z ) | 2 = 4 9 sin 2 [ j( γ 2 γ 4 )z 2 ] for Δ 1 < Δ 2 < Δ 3 | A 2 ( z ) | 2 = 1 9 + 8 9 cos 2 [ j( γ 1 γ 4 )z 2 ], | A 4 ( z ) | 2 = | A 6 ( z ) | 2 = 4 9 sin 2 [ j( γ 1 γ 4 )z 2 ] for Δ 3 < Δ 2 < Δ 1
c2 ={ π/| j( γ 2 γ 5 ) | for Δ 2 < Δ 1 , Δ 3 π/| j( γ 1 γ 3 ) | for Δ 2 > Δ 1 , Δ 3 π/| j( γ 2 γ 4 ) | for Δ 1 < Δ 2 < Δ 3 π/| j( γ 1 γ 4 ) | for Δ 3 < Δ 2 < Δ 1
| A 2 ( c2 ) | 2 = 1/9.
E(0)= [ E 1 (0) E 2 (0) E 3 (0) E 4 (0) E 5 (0) E 6 (0) E 7 (0) ] H ,
E(z)= e Rz E(0)=TE(0).
y p = E p (z)+ υ p , p=1,2,3,4,5,6,7
E ^ (0)= T 1 y= W H y=E(0)+ W H υ.
W (n+1) = W (n) +μ y (n) ( E (n) (0) ( W (n) ) H y (n) ) H = W (n) +μ y (n) e (n) H
s p = | E p (z) | 2 + υ p = | t p E(0) | 2 + υ p
E ^ (0)=arg min x { 0,1 } 7 ( s | Tx | 2 ) T ( s | Tx | 2 ).
γ 1 =j( β 2 C 24 ), γ 2 =j( β 3 C 35 ) , γ 3 =j( α 22 + α 33 + S ¯ ) , γ 4 =j( α 22 + α 33 S ¯ )
α 22 = β 2 + C 24 2 , α 33 = β 3 + C 35 2 , S ¯ = ( α 22 α 33 ) 2 +4 C 23 C 32
A 2 ( z )= 1 2 exp[ j( α 22 + α 33 β 2 )z ][ cos( S ¯ z )j α 22 α 33 S ¯ sin( S ¯ z ) ]+ 1 2 exp( j C 24 z ),
A 4 ( z )= 1 2 exp[ j( α 22 + α 33 β 2 )z ][ cos( S ¯ z )j α 22 α 33 S ¯ sin( S ¯ z ) ] 1 2 exp( j C 24 z ),
A 3 ( z )= A 5 ( z )=j C 32 S ¯ exp[ j( α 22 + α 33 β 3 )z ]sin( S ¯ z ),
| A 2 ( z ) | 2 = 1 4 + 1 4 cos 2 ( S ¯ z )+ 1 2 cos( S ¯ z )cos[ ( α 22 + α 33 + C 24 β 2 )z ] + α 22 α 33 4 S ¯ sin( S ¯ z ){ α 22 α 33 S ¯ sin( S ¯ z )2sin[ ( α 22 + α 33 + C 24 β 2 )z ] },
| A 4 ( z ) | 2 = 1 4 + 1 4 cos 2 ( S ¯ z ) 1 2 cos( S ¯ z )cos[ ( α 22 + α 33 + C 24 β 2 )z ] + α 22 α 33 4 S ¯ sin( S ¯ z ){ α 22 α 33 S ¯ sin( S ¯ z )+2sin[ ( α 22 + α 33 + C 24 β 2 )z ] },
| A 3 ( z ) | 2 = | A 5 ( z ) | 2 = ( C 32 S ¯ ) 2 sin 2 ( S ¯ z ).
| A 2 ( z ) | 2 = cos 2 [ ( α 22 α 33 )( C 24 α 22 α 33 +ξ )z ]ξsin( S ¯ z ){ sin( S ¯ z )sin[ ( α 22 + α 33 + C 24 β 2 )z ] } cos 2 [ ( α 22 α 33 )( C 24 α 22 α 33 +ξ )z ]
| A 3 ( z ) | 2 = | A 5 ( z ) | 2 0.
| A 2 ( z ) | 2 = cos 2 ( j( γ 1 γ 3 ) 2 z ), | A 4 ( z ) | 2 = sin 2 ( j( γ 1 γ 3 ) 2 z ) for Δ 2 > Δ 3 | A 2 ( z ) | 2 = cos 2 ( j( γ 1 γ 4 ) 2 z ), | A 4 ( z ) | 2 = sin 2 ( j( γ 1 γ 4 ) 2 z ) for Δ 2 < Δ 3
c2 ={ π/| j( γ 1 γ 3 ) | for Δ 2 > Δ 3 π/| j( γ 1 γ 4 ) | for Δ 2 < Δ 3
| A 2 ( c1 ) | 2 =0.

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