Abstract

We systematically investigate the intercore coupling in triple-core photonic crystal fibers (PCFs). Coupled-mode equations are developed and solved analytically in the linear approximation. We derive the eigenmodes in triple-core PCFs, in particular we model the decoupling mode and test its stability against perturbations. The coupling coefficients and the coupling length are determined experimentally and found to agree extremely well with the calculated results. We describe the propagation of light in triple-core PCFs for different launching conditions.

© 2008 Optical Society of America

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    [Crossref]

2006 (5)

P. St. J. Russell, “Photonic-crystal fibers,” J. Lumin. 24, 4729-4749 (2006).

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135-1184 (2006).
[Crossref]

D. M. Taylor, C. R. Bennett, T. J. Shepherd, L. F. Michaille, M. D. Nielsen, and H. R. Simonsen, “Demonstration of multi-core photonic crystal fibre in an optical interconnect,” Electron. Lett. 42, 331-332 (2006).
[Crossref]

L. Michaille, C. R. Bennett, D. M. Taylor, and T. J. Shepherd, “Multi-core photonic crystal fibers for high power lasers and amplifiers,” Proc. SPIE 6102, 61020W (2006).
[Crossref]

S. K. Varshney, N. J. Florous, K. Saitoh, and M. Koshiba, “The impact of elliptical deformations for optimizing the performance of dual-core fluorine-doped photonic crystal fiber couplers,” Opt. Express 14, 1982-1995 (2006).
[Crossref] [PubMed]

2005 (4)

2004 (2)

Y. Jiang, Z. Shi, B. Howley, and R. T. Chen, “Highly dispersive photonic crystal fibers for true-time-delay modules of an x-band phased array antenna,” Proc. SPIE 5360, 253-258 (2004).
[Crossref]

Y. Huo, P. K. Cheo, and G. G. King, “Fundamental mode operation of a 19-core phase-locked Yb-doped fiber amplifier,” Opt. Express 12, 6230-6239 (2004).
[Crossref] [PubMed]

2003 (5)

K. Saitoh, Y. Sato, and M. Koshiba, “Coupling characteristics of dual-core photonic crystal fiber couplers,” Opt. Express 11, 3188-3195 (2003).
[Crossref] [PubMed]

W. J. Wadsworth, R. M. Percival, G. Bouwmans, J. C. Knight, and P. St. J. Russell, “High power air-clad photonic crystal fibre laser,” Opt. Express 11, 48-53 (2003).
[Crossref] [PubMed]

L. P. Shen, W.-P. Huang, G. X. Chen, and S. S. Jian, “Design and optimization of photonic crystal fibers for broad-band dispersion compensation,” IEEE Photon. Technol. Lett. 15, 540-542 (2003).
[Crossref]

A. I. Siahlo, L. K. Oxenlowe, K. S. Berg, A. T. Clausen, P. A. Andersen, C. Peucheret, A. Tersigni, P. Jeppesen, K. P. Hansen, and J. R. Folkenberg, “A high-speed demultiplexer based on a nonlinear optical loop mirror with a photonic crystal fiber,” IEEE Photon. Technol. Lett. 15, 1147-1149 (2003).
[Crossref]

M. A. van Eijkelenborg, A. Argyros, G. Barton, I. M. Bassett, M. Fellew, G. Henry, N. A. Issa, M. C. J. Large, S. Manos, W. Padden, L. Poladian, and J. Zagari, “Recent progress in microstructured polymer optical fibre fabrication and characterization,” Opt. Fiber Technol. 9, 199-209 (2003).
[Crossref]

2002 (2)

2001 (1)

W. N. MacPherson, M. J. Gander, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, B. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Remotely addressed optical fibre curvature sensor using multicore photonic crystal fibre,” Opt. Commun. 193, 97-104 (2001).
[Crossref]

2000 (5)

B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, and A. H. Greenaway, “Experimental study of dual-core photonic crystal fibre,” Electron. Lett. 36, 1358-1359 (2000).
[Crossref]

M. Wrage, P. Glas, M. Leitner, T. Sandrock, N. N. Elkin, A. P. Napartovich, and A. G. Sukharev, “Experimental and numerical determination of coupling constant in a multicore fiber,” Opt. Commun. 175, 97-102 (2000).
[Crossref]

T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. 18, 50-56 (2000).
[Crossref]

B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler, and G. L. Burdge, “Cladding-mode-resonances in air-silica microstructure optical fibers,” J. Lightwave Technol. 18, 1084-1100 (2000).
[Crossref]

P. M. Blanchard, J. G. Burnett, G. R. G. Erry, A. H. Greenaway, P. Harrison, B. Mangan, J. C. Knight, P. St. J. Russell, M. J. Gander, R. McBride, and J. D. C. Jones, “Two-dimensional bend sensing with a single multicore optical fibre,” Smart Mater. Struct. 9, 132-140 (2000).
[Crossref]

1999 (1)

1998 (1)

1996 (1)

1986 (1)

R. Tewari and K. Thyagarajan, “Analysis of tunable single-mode fiber directional couplers using simple and accurate relations,” J. Lightwave Technol. 4, 386-390 (1986).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic, 2001).

Alic, N.

R. Jiang, R. Saperstein, N. Alic, M. Nezhad, C. McKinstrie, J. Ford, Y. Fainman, and S. Radic, “375 THz parametric translation of modulated signal from 1550 nm to visible band,” in Optical Fiber Communication (Optical Society of America, 2006), postdeadline paper 16.

Andersen, P. A.

A. I. Siahlo, L. K. Oxenlowe, K. S. Berg, A. T. Clausen, P. A. Andersen, C. Peucheret, A. Tersigni, P. Jeppesen, K. P. Hansen, and J. R. Folkenberg, “A high-speed demultiplexer based on a nonlinear optical loop mirror with a photonic crystal fiber,” IEEE Photon. Technol. Lett. 15, 1147-1149 (2003).
[Crossref]

Andrés, M. V.

Andrés, P.

Argyros, A.

M. A. van Eijkelenborg, A. Argyros, G. Barton, I. M. Bassett, M. Fellew, G. Henry, N. A. Issa, M. C. J. Large, S. Manos, W. Padden, L. Poladian, and J. Zagari, “Recent progress in microstructured polymer optical fibre fabrication and characterization,” Opt. Fiber Technol. 9, 199-209 (2003).
[Crossref]

Atkin, D. M.

Barton, G.

M. A. van Eijkelenborg, A. Argyros, G. Barton, I. M. Bassett, M. Fellew, G. Henry, N. A. Issa, M. C. J. Large, S. Manos, W. Padden, L. Poladian, and J. Zagari, “Recent progress in microstructured polymer optical fibre fabrication and characterization,” Opt. Fiber Technol. 9, 199-209 (2003).
[Crossref]

Bassett, I. M.

M. A. van Eijkelenborg, A. Argyros, G. Barton, I. M. Bassett, M. Fellew, G. Henry, N. A. Issa, M. C. J. Large, S. Manos, W. Padden, L. Poladian, and J. Zagari, “Recent progress in microstructured polymer optical fibre fabrication and characterization,” Opt. Fiber Technol. 9, 199-209 (2003).
[Crossref]

Bassi, P.

Beach, R.

R. Beach and L. Zapata, “Tactical laser weapons for defense SI,” Laboratory Directed Research and Development final report (U.S. Department of Energy, 2002).

Bellanca, G.

Bennett, C. R.

L. Michaille, C. R. Bennett, D. M. Taylor, and T. J. Shepherd, “Multi-core photonic crystal fibers for high power lasers and amplifiers,” Proc. SPIE 6102, 61020W (2006).
[Crossref]

D. M. Taylor, C. R. Bennett, T. J. Shepherd, L. F. Michaille, M. D. Nielsen, and H. R. Simonsen, “Demonstration of multi-core photonic crystal fibre in an optical interconnect,” Electron. Lett. 42, 331-332 (2006).
[Crossref]

Bennett, P. J.

Berg, K. S.

A. I. Siahlo, L. K. Oxenlowe, K. S. Berg, A. T. Clausen, P. A. Andersen, C. Peucheret, A. Tersigni, P. Jeppesen, K. P. Hansen, and J. R. Folkenberg, “A high-speed demultiplexer based on a nonlinear optical loop mirror with a photonic crystal fiber,” IEEE Photon. Technol. Lett. 15, 1147-1149 (2003).
[Crossref]

Birks, T. A.

W. N. MacPherson, M. J. Gander, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, B. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Remotely addressed optical fibre curvature sensor using multicore photonic crystal fibre,” Opt. Commun. 193, 97-104 (2001).
[Crossref]

B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, and A. H. Greenaway, “Experimental study of dual-core photonic crystal fibre,” Electron. Lett. 36, 1358-1359 (2000).
[Crossref]

J. C. Knight, T. A. Birks, P. St. J. Russell, and J. P. de Sandro, “Properties of photonic crystal fiber and the effective index model,” J. Opt. Soc. Am. A 15, 748-752 (1998).
[Crossref]

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547-1549 (1996).
[Crossref] [PubMed]

G. Kakarantzas, B. J. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Directional coupling in a twin core photonic crystal fiber using heat treatment,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2001), paper JTuD2.

Blanchard, P. M.

W. N. MacPherson, M. J. Gander, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, B. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Remotely addressed optical fibre curvature sensor using multicore photonic crystal fibre,” Opt. Commun. 193, 97-104 (2001).
[Crossref]

P. M. Blanchard, J. G. Burnett, G. R. G. Erry, A. H. Greenaway, P. Harrison, B. Mangan, J. C. Knight, P. St. J. Russell, M. J. Gander, R. McBride, and J. D. C. Jones, “Two-dimensional bend sensing with a single multicore optical fibre,” Smart Mater. Struct. 9, 132-140 (2000).
[Crossref]

Botten, L. C.

Bouwmans, G.

Broderick, N. G. R.

Burdge, G. L.

Burnett, J. G.

W. N. MacPherson, M. J. Gander, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, B. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Remotely addressed optical fibre curvature sensor using multicore photonic crystal fibre,” Opt. Commun. 193, 97-104 (2001).
[Crossref]

P. M. Blanchard, J. G. Burnett, G. R. G. Erry, A. H. Greenaway, P. Harrison, B. Mangan, J. C. Knight, P. St. J. Russell, M. J. Gander, R. McBride, and J. D. C. Jones, “Two-dimensional bend sensing with a single multicore optical fibre,” Smart Mater. Struct. 9, 132-140 (2000).
[Crossref]

Chen, A. Y. H.

Chen, G. X.

L. P. Shen, W.-P. Huang, G. X. Chen, and S. S. Jian, “Design and optimization of photonic crystal fibers for broad-band dispersion compensation,” IEEE Photon. Technol. Lett. 15, 540-542 (2003).
[Crossref]

Chen, R. T.

Y. Jiang, Z. Shi, B. Howley, and R. T. Chen, “Highly dispersive photonic crystal fibers for true-time-delay modules of an x-band phased array antenna,” Proc. SPIE 5360, 253-258 (2004).
[Crossref]

Cheo, P. K.

Clausen, A. T.

A. I. Siahlo, L. K. Oxenlowe, K. S. Berg, A. T. Clausen, P. A. Andersen, C. Peucheret, A. Tersigni, P. Jeppesen, K. P. Hansen, and J. R. Folkenberg, “A high-speed demultiplexer based on a nonlinear optical loop mirror with a photonic crystal fiber,” IEEE Photon. Technol. Lett. 15, 1147-1149 (2003).
[Crossref]

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135-1184 (2006).
[Crossref]

de Sandro, J. P.

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135-1184 (2006).
[Crossref]

Duligall, J.

Eggleton, B. J.

Elkin, N. N.

M. Wrage, P. Glas, M. Leitner, T. Sandrock, N. N. Elkin, A. P. Napartovich, and A. G. Sukharev, “Experimental and numerical determination of coupling constant in a multicore fiber,” Opt. Commun. 175, 97-102 (2000).
[Crossref]

Erry, G. R. G.

P. M. Blanchard, J. G. Burnett, G. R. G. Erry, A. H. Greenaway, P. Harrison, B. Mangan, J. C. Knight, P. St. J. Russell, M. J. Gander, R. McBride, and J. D. C. Jones, “Two-dimensional bend sensing with a single multicore optical fibre,” Smart Mater. Struct. 9, 132-140 (2000).
[Crossref]

Fainman, Y.

R. Jiang, R. Saperstein, N. Alic, M. Nezhad, C. McKinstrie, J. Ford, Y. Fainman, and S. Radic, “375 THz parametric translation of modulated signal from 1550 nm to visible band,” in Optical Fiber Communication (Optical Society of America, 2006), postdeadline paper 16.

Fellew, M.

M. A. van Eijkelenborg, A. Argyros, G. Barton, I. M. Bassett, M. Fellew, G. Henry, N. A. Issa, M. C. J. Large, S. Manos, W. Padden, L. Poladian, and J. Zagari, “Recent progress in microstructured polymer optical fibre fabrication and characterization,” Opt. Fiber Technol. 9, 199-209 (2003).
[Crossref]

Ferrando, A.

Florous, N. J.

Fogli, F.

Folkenberg, J. R.

A. I. Siahlo, L. K. Oxenlowe, K. S. Berg, A. T. Clausen, P. A. Andersen, C. Peucheret, A. Tersigni, P. Jeppesen, K. P. Hansen, and J. R. Folkenberg, “A high-speed demultiplexer based on a nonlinear optical loop mirror with a photonic crystal fiber,” IEEE Photon. Technol. Lett. 15, 1147-1149 (2003).
[Crossref]

Ford, J.

R. Jiang, R. Saperstein, N. Alic, M. Nezhad, C. McKinstrie, J. Ford, Y. Fainman, and S. Radic, “375 THz parametric translation of modulated signal from 1550 nm to visible band,” in Optical Fiber Communication (Optical Society of America, 2006), postdeadline paper 16.

Fulconis, J.

Gallagher, M. T.

J. A. West, N. Venkataramam, C. M. Smith, and M. T. Gallagher, “Photonic crystal fibers,” in 27th European Conference on Optical Comunication (IEEE, 2001), paper Th. A. 2.2.

Gander, M. J.

W. N. MacPherson, M. J. Gander, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, B. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Remotely addressed optical fibre curvature sensor using multicore photonic crystal fibre,” Opt. Commun. 193, 97-104 (2001).
[Crossref]

P. M. Blanchard, J. G. Burnett, G. R. G. Erry, A. H. Greenaway, P. Harrison, B. Mangan, J. C. Knight, P. St. J. Russell, M. J. Gander, R. McBride, and J. D. C. Jones, “Two-dimensional bend sensing with a single multicore optical fibre,” Smart Mater. Struct. 9, 132-140 (2000).
[Crossref]

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135-1184 (2006).
[Crossref]

Glas, P.

M. Wrage, P. Glas, M. Leitner, T. Sandrock, N. N. Elkin, A. P. Napartovich, and A. G. Sukharev, “Experimental and numerical determination of coupling constant in a multicore fiber,” Opt. Commun. 175, 97-102 (2000).
[Crossref]

Greenaway, A. H.

W. N. MacPherson, M. J. Gander, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, B. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Remotely addressed optical fibre curvature sensor using multicore photonic crystal fibre,” Opt. Commun. 193, 97-104 (2001).
[Crossref]

B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, and A. H. Greenaway, “Experimental study of dual-core photonic crystal fibre,” Electron. Lett. 36, 1358-1359 (2000).
[Crossref]

P. M. Blanchard, J. G. Burnett, G. R. G. Erry, A. H. Greenaway, P. Harrison, B. Mangan, J. C. Knight, P. St. J. Russell, M. J. Gander, R. McBride, and J. D. C. Jones, “Two-dimensional bend sensing with a single multicore optical fibre,” Smart Mater. Struct. 9, 132-140 (2000).
[Crossref]

Ha, S. W.

Hansen, K. P.

A. I. Siahlo, L. K. Oxenlowe, K. S. Berg, A. T. Clausen, P. A. Andersen, C. Peucheret, A. Tersigni, P. Jeppesen, K. P. Hansen, and J. R. Folkenberg, “A high-speed demultiplexer based on a nonlinear optical loop mirror with a photonic crystal fiber,” IEEE Photon. Technol. Lett. 15, 1147-1149 (2003).
[Crossref]

Harrison, P.

P. M. Blanchard, J. G. Burnett, G. R. G. Erry, A. H. Greenaway, P. Harrison, B. Mangan, J. C. Knight, P. St. J. Russell, M. J. Gander, R. McBride, and J. D. C. Jones, “Two-dimensional bend sensing with a single multicore optical fibre,” Smart Mater. Struct. 9, 132-140 (2000).
[Crossref]

Harvey, J. D.

Henry, G.

M. A. van Eijkelenborg, A. Argyros, G. Barton, I. M. Bassett, M. Fellew, G. Henry, N. A. Issa, M. C. J. Large, S. Manos, W. Padden, L. Poladian, and J. Zagari, “Recent progress in microstructured polymer optical fibre fabrication and characterization,” Opt. Fiber Technol. 9, 199-209 (2003).
[Crossref]

Howley, B.

Y. Jiang, Z. Shi, B. Howley, and R. T. Chen, “Highly dispersive photonic crystal fibers for true-time-delay modules of an x-band phased array antenna,” Proc. SPIE 5360, 253-258 (2004).
[Crossref]

Huang, W.-P.

L. P. Shen, W.-P. Huang, G. X. Chen, and S. S. Jian, “Design and optimization of photonic crystal fibers for broad-band dispersion compensation,” IEEE Photon. Technol. Lett. 15, 540-542 (2003).
[Crossref]

Huo, Y.

Issa, N. A.

M. A. van Eijkelenborg, A. Argyros, G. Barton, I. M. Bassett, M. Fellew, G. Henry, N. A. Issa, M. C. J. Large, S. Manos, W. Padden, L. Poladian, and J. Zagari, “Recent progress in microstructured polymer optical fibre fabrication and characterization,” Opt. Fiber Technol. 9, 199-209 (2003).
[Crossref]

Jeppesen, P.

A. I. Siahlo, L. K. Oxenlowe, K. S. Berg, A. T. Clausen, P. A. Andersen, C. Peucheret, A. Tersigni, P. Jeppesen, K. P. Hansen, and J. R. Folkenberg, “A high-speed demultiplexer based on a nonlinear optical loop mirror with a photonic crystal fiber,” IEEE Photon. Technol. Lett. 15, 1147-1149 (2003).
[Crossref]

Jian, S. S.

L. P. Shen, W.-P. Huang, G. X. Chen, and S. S. Jian, “Design and optimization of photonic crystal fibers for broad-band dispersion compensation,” IEEE Photon. Technol. Lett. 15, 540-542 (2003).
[Crossref]

Jiang, R.

R. Jiang, R. Saperstein, N. Alic, M. Nezhad, C. McKinstrie, J. Ford, Y. Fainman, and S. Radic, “375 THz parametric translation of modulated signal from 1550 nm to visible band,” in Optical Fiber Communication (Optical Society of America, 2006), postdeadline paper 16.

Jiang, Y.

Y. Jiang, Z. Shi, B. Howley, and R. T. Chen, “Highly dispersive photonic crystal fibers for true-time-delay modules of an x-band phased array antenna,” Proc. SPIE 5360, 253-258 (2004).
[Crossref]

Joly, N.

Jones, J. D. C.

W. N. MacPherson, M. J. Gander, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, B. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Remotely addressed optical fibre curvature sensor using multicore photonic crystal fibre,” Opt. Commun. 193, 97-104 (2001).
[Crossref]

P. M. Blanchard, J. G. Burnett, G. R. G. Erry, A. H. Greenaway, P. Harrison, B. Mangan, J. C. Knight, P. St. J. Russell, M. J. Gander, R. McBride, and J. D. C. Jones, “Two-dimensional bend sensing with a single multicore optical fibre,” Smart Mater. Struct. 9, 132-140 (2000).
[Crossref]

Kakarantzas, G.

G. Kakarantzas, B. J. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Directional coupling in a twin core photonic crystal fiber using heat treatment,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2001), paper JTuD2.

Kerbage, C.

King, G. G.

Knight, J. C.

W. J. Wadsworth, R. M. Percival, G. Bouwmans, J. C. Knight, and P. St. J. Russell, “High power air-clad photonic crystal fibre laser,” Opt. Express 11, 48-53 (2003).
[Crossref] [PubMed]

W. N. MacPherson, M. J. Gander, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, B. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Remotely addressed optical fibre curvature sensor using multicore photonic crystal fibre,” Opt. Commun. 193, 97-104 (2001).
[Crossref]

P. M. Blanchard, J. G. Burnett, G. R. G. Erry, A. H. Greenaway, P. Harrison, B. Mangan, J. C. Knight, P. St. J. Russell, M. J. Gander, R. McBride, and J. D. C. Jones, “Two-dimensional bend sensing with a single multicore optical fibre,” Smart Mater. Struct. 9, 132-140 (2000).
[Crossref]

B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, and A. H. Greenaway, “Experimental study of dual-core photonic crystal fibre,” Electron. Lett. 36, 1358-1359 (2000).
[Crossref]

J. C. Knight, T. A. Birks, P. St. J. Russell, and J. P. de Sandro, “Properties of photonic crystal fiber and the effective index model,” J. Opt. Soc. Am. A 15, 748-752 (1998).
[Crossref]

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547-1549 (1996).
[Crossref] [PubMed]

G. Kakarantzas, B. J. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Directional coupling in a twin core photonic crystal fiber using heat treatment,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2001), paper JTuD2.

Koshiba, M.

Kruhlak, R. J.

Kuhlmey, B. T.

Lægsgaard, J.

Large, M. C. J.

M. A. van Eijkelenborg, A. Argyros, G. Barton, I. M. Bassett, M. Fellew, G. Henry, N. A. Issa, M. C. J. Large, S. Manos, W. Padden, L. Poladian, and J. Zagari, “Recent progress in microstructured polymer optical fibre fabrication and characterization,” Opt. Fiber Technol. 9, 199-209 (2003).
[Crossref]

Leitner, M.

M. Wrage, P. Glas, M. Leitner, T. Sandrock, N. N. Elkin, A. P. Napartovich, and A. G. Sukharev, “Experimental and numerical determination of coupling constant in a multicore fiber,” Opt. Commun. 175, 97-102 (2000).
[Crossref]

Leonhardt, R.

MacPherson, W. N.

W. N. MacPherson, M. J. Gander, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, B. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Remotely addressed optical fibre curvature sensor using multicore photonic crystal fibre,” Opt. Commun. 193, 97-104 (2001).
[Crossref]

Mangan, B.

W. N. MacPherson, M. J. Gander, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, B. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Remotely addressed optical fibre curvature sensor using multicore photonic crystal fibre,” Opt. Commun. 193, 97-104 (2001).
[Crossref]

P. M. Blanchard, J. G. Burnett, G. R. G. Erry, A. H. Greenaway, P. Harrison, B. Mangan, J. C. Knight, P. St. J. Russell, M. J. Gander, R. McBride, and J. D. C. Jones, “Two-dimensional bend sensing with a single multicore optical fibre,” Smart Mater. Struct. 9, 132-140 (2000).
[Crossref]

Mangan, B. J.

B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, and A. H. Greenaway, “Experimental study of dual-core photonic crystal fibre,” Electron. Lett. 36, 1358-1359 (2000).
[Crossref]

G. Kakarantzas, B. J. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Directional coupling in a twin core photonic crystal fiber using heat treatment,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2001), paper JTuD2.

Manos, S.

M. A. van Eijkelenborg, A. Argyros, G. Barton, I. M. Bassett, M. Fellew, G. Henry, N. A. Issa, M. C. J. Large, S. Manos, W. Padden, L. Poladian, and J. Zagari, “Recent progress in microstructured polymer optical fibre fabrication and characterization,” Opt. Fiber Technol. 9, 199-209 (2003).
[Crossref]

Martijn de Sterke, C.

Maystre, D.

McBride, R.

W. N. MacPherson, M. J. Gander, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, B. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Remotely addressed optical fibre curvature sensor using multicore photonic crystal fibre,” Opt. Commun. 193, 97-104 (2001).
[Crossref]

P. M. Blanchard, J. G. Burnett, G. R. G. Erry, A. H. Greenaway, P. Harrison, B. Mangan, J. C. Knight, P. St. J. Russell, M. J. Gander, R. McBride, and J. D. C. Jones, “Two-dimensional bend sensing with a single multicore optical fibre,” Smart Mater. Struct. 9, 132-140 (2000).
[Crossref]

McKinstrie, C.

R. Jiang, R. Saperstein, N. Alic, M. Nezhad, C. McKinstrie, J. Ford, Y. Fainman, and S. Radic, “375 THz parametric translation of modulated signal from 1550 nm to visible band,” in Optical Fiber Communication (Optical Society of America, 2006), postdeadline paper 16.

McPhedran, R. C.

Michaille, L.

L. Michaille, C. R. Bennett, D. M. Taylor, and T. J. Shepherd, “Multi-core photonic crystal fibers for high power lasers and amplifiers,” Proc. SPIE 6102, 61020W (2006).
[Crossref]

Michaille, L. F.

D. M. Taylor, C. R. Bennett, T. J. Shepherd, L. F. Michaille, M. D. Nielsen, and H. R. Simonsen, “Demonstration of multi-core photonic crystal fibre in an optical interconnect,” Electron. Lett. 42, 331-332 (2006).
[Crossref]

Miret, J. J.

Monro, T. M.

Murdoch, S. G.

Napartovich, A. P.

M. Wrage, P. Glas, M. Leitner, T. Sandrock, N. N. Elkin, A. P. Napartovich, and A. G. Sukharev, “Experimental and numerical determination of coupling constant in a multicore fiber,” Opt. Commun. 175, 97-102 (2000).
[Crossref]

Nezhad, M.

R. Jiang, R. Saperstein, N. Alic, M. Nezhad, C. McKinstrie, J. Ford, Y. Fainman, and S. Radic, “375 THz parametric translation of modulated signal from 1550 nm to visible band,” in Optical Fiber Communication (Optical Society of America, 2006), postdeadline paper 16.

Nielsen, M. D.

D. M. Taylor, C. R. Bennett, T. J. Shepherd, L. F. Michaille, M. D. Nielsen, and H. R. Simonsen, “Demonstration of multi-core photonic crystal fibre in an optical interconnect,” Electron. Lett. 42, 331-332 (2006).
[Crossref]

Oxenlowe, L. K.

A. I. Siahlo, L. K. Oxenlowe, K. S. Berg, A. T. Clausen, P. A. Andersen, C. Peucheret, A. Tersigni, P. Jeppesen, K. P. Hansen, and J. R. Folkenberg, “A high-speed demultiplexer based on a nonlinear optical loop mirror with a photonic crystal fiber,” IEEE Photon. Technol. Lett. 15, 1147-1149 (2003).
[Crossref]

Padden, W.

M. A. van Eijkelenborg, A. Argyros, G. Barton, I. M. Bassett, M. Fellew, G. Henry, N. A. Issa, M. C. J. Large, S. Manos, W. Padden, L. Poladian, and J. Zagari, “Recent progress in microstructured polymer optical fibre fabrication and characterization,” Opt. Fiber Technol. 9, 199-209 (2003).
[Crossref]

Percival, R. M.

Peucheret, C.

A. I. Siahlo, L. K. Oxenlowe, K. S. Berg, A. T. Clausen, P. A. Andersen, C. Peucheret, A. Tersigni, P. Jeppesen, K. P. Hansen, and J. R. Folkenberg, “A high-speed demultiplexer based on a nonlinear optical loop mirror with a photonic crystal fiber,” IEEE Photon. Technol. Lett. 15, 1147-1149 (2003).
[Crossref]

Poladian, L.

M. A. van Eijkelenborg, A. Argyros, G. Barton, I. M. Bassett, M. Fellew, G. Henry, N. A. Issa, M. C. J. Large, S. Manos, W. Padden, L. Poladian, and J. Zagari, “Recent progress in microstructured polymer optical fibre fabrication and characterization,” Opt. Fiber Technol. 9, 199-209 (2003).
[Crossref]

Radic, S.

R. Jiang, R. Saperstein, N. Alic, M. Nezhad, C. McKinstrie, J. Ford, Y. Fainman, and S. Radic, “375 THz parametric translation of modulated signal from 1550 nm to visible band,” in Optical Fiber Communication (Optical Society of America, 2006), postdeadline paper 16.

Rarity, J. G.

Renversez, G.

Richardson, D. J.

Russell, P. St. J.

P. St. J. Russell, “Photonic-crystal fibers,” J. Lumin. 24, 4729-4749 (2006).

J. G. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth and P. St. J. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13, 534-544 (2005).
[Crossref] [PubMed]

W. J. Wadsworth, R. M. Percival, G. Bouwmans, J. C. Knight, and P. St. J. Russell, “High power air-clad photonic crystal fibre laser,” Opt. Express 11, 48-53 (2003).
[Crossref] [PubMed]

W. N. MacPherson, M. J. Gander, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, B. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Remotely addressed optical fibre curvature sensor using multicore photonic crystal fibre,” Opt. Commun. 193, 97-104 (2001).
[Crossref]

P. M. Blanchard, J. G. Burnett, G. R. G. Erry, A. H. Greenaway, P. Harrison, B. Mangan, J. C. Knight, P. St. J. Russell, M. J. Gander, R. McBride, and J. D. C. Jones, “Two-dimensional bend sensing with a single multicore optical fibre,” Smart Mater. Struct. 9, 132-140 (2000).
[Crossref]

B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, and A. H. Greenaway, “Experimental study of dual-core photonic crystal fibre,” Electron. Lett. 36, 1358-1359 (2000).
[Crossref]

J. C. Knight, T. A. Birks, P. St. J. Russell, and J. P. de Sandro, “Properties of photonic crystal fiber and the effective index model,” J. Opt. Soc. Am. A 15, 748-752 (1998).
[Crossref]

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547-1549 (1996).
[Crossref] [PubMed]

G. Kakarantzas, B. J. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Directional coupling in a twin core photonic crystal fiber using heat treatment,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2001), paper JTuD2.

Saccomandi, L.

Saitoh, K.

Sandrock, T.

M. Wrage, P. Glas, M. Leitner, T. Sandrock, N. N. Elkin, A. P. Napartovich, and A. G. Sukharev, “Experimental and numerical determination of coupling constant in a multicore fiber,” Opt. Commun. 175, 97-102 (2000).
[Crossref]

Saperstein, R.

R. Jiang, R. Saperstein, N. Alic, M. Nezhad, C. McKinstrie, J. Ford, Y. Fainman, and S. Radic, “375 THz parametric translation of modulated signal from 1550 nm to visible band,” in Optical Fiber Communication (Optical Society of America, 2006), postdeadline paper 16.

Sato, Y.

Shen, L. P.

L. P. Shen, W.-P. Huang, G. X. Chen, and S. S. Jian, “Design and optimization of photonic crystal fibers for broad-band dispersion compensation,” IEEE Photon. Technol. Lett. 15, 540-542 (2003).
[Crossref]

Shepherd, T. J.

D. M. Taylor, C. R. Bennett, T. J. Shepherd, L. F. Michaille, M. D. Nielsen, and H. R. Simonsen, “Demonstration of multi-core photonic crystal fibre in an optical interconnect,” Electron. Lett. 42, 331-332 (2006).
[Crossref]

L. Michaille, C. R. Bennett, D. M. Taylor, and T. J. Shepherd, “Multi-core photonic crystal fibers for high power lasers and amplifiers,” Proc. SPIE 6102, 61020W (2006).
[Crossref]

Shi, Z.

Y. Jiang, Z. Shi, B. Howley, and R. T. Chen, “Highly dispersive photonic crystal fibers for true-time-delay modules of an x-band phased array antenna,” Proc. SPIE 5360, 253-258 (2004).
[Crossref]

Siahlo, A. I.

A. I. Siahlo, L. K. Oxenlowe, K. S. Berg, A. T. Clausen, P. A. Andersen, C. Peucheret, A. Tersigni, P. Jeppesen, K. P. Hansen, and J. R. Folkenberg, “A high-speed demultiplexer based on a nonlinear optical loop mirror with a photonic crystal fiber,” IEEE Photon. Technol. Lett. 15, 1147-1149 (2003).
[Crossref]

Silvestre, E.

Simonsen, H. R.

D. M. Taylor, C. R. Bennett, T. J. Shepherd, L. F. Michaille, M. D. Nielsen, and H. R. Simonsen, “Demonstration of multi-core photonic crystal fibre in an optical interconnect,” Electron. Lett. 42, 331-332 (2006).
[Crossref]

Skorobogatiy, M.

Smith, C. M.

J. A. West, N. Venkataramam, C. M. Smith, and M. T. Gallagher, “Photonic crystal fibers,” in 27th European Conference on Optical Comunication (IEEE, 2001), paper Th. A. 2.2.

Sukharev, A. G.

M. Wrage, P. Glas, M. Leitner, T. Sandrock, N. N. Elkin, A. P. Napartovich, and A. G. Sukharev, “Experimental and numerical determination of coupling constant in a multicore fiber,” Opt. Commun. 175, 97-102 (2000).
[Crossref]

Taylor, D. M.

L. Michaille, C. R. Bennett, D. M. Taylor, and T. J. Shepherd, “Multi-core photonic crystal fibers for high power lasers and amplifiers,” Proc. SPIE 6102, 61020W (2006).
[Crossref]

D. M. Taylor, C. R. Bennett, T. J. Shepherd, L. F. Michaille, M. D. Nielsen, and H. R. Simonsen, “Demonstration of multi-core photonic crystal fibre in an optical interconnect,” Electron. Lett. 42, 331-332 (2006).
[Crossref]

Tersigni, A.

A. I. Siahlo, L. K. Oxenlowe, K. S. Berg, A. T. Clausen, P. A. Andersen, C. Peucheret, A. Tersigni, P. Jeppesen, K. P. Hansen, and J. R. Folkenberg, “A high-speed demultiplexer based on a nonlinear optical loop mirror with a photonic crystal fiber,” IEEE Photon. Technol. Lett. 15, 1147-1149 (2003).
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R. Tewari and K. Thyagarajan, “Analysis of tunable single-mode fiber directional couplers using simple and accurate relations,” J. Lightwave Technol. 4, 386-390 (1986).
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I. Velchev and J. Toulouse, “Directional coupling and switching in multi-core microstructure fibers,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2004), paper CTuV1.

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van Eijkelenborg, M. A.

M. A. van Eijkelenborg, A. Argyros, G. Barton, I. M. Bassett, M. Fellew, G. Henry, N. A. Issa, M. C. J. Large, S. Manos, W. Padden, L. Poladian, and J. Zagari, “Recent progress in microstructured polymer optical fibre fabrication and characterization,” Opt. Fiber Technol. 9, 199-209 (2003).
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Velchev, I.

I. Velchev and J. Toulouse, “Directional coupling and switching in multi-core microstructure fibers,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2004), paper CTuV1.

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J. A. West, N. Venkataramam, C. M. Smith, and M. T. Gallagher, “Photonic crystal fibers,” in 27th European Conference on Optical Comunication (IEEE, 2001), paper Th. A. 2.2.

Wadsworth, W. J.

West, J. A.

J. A. West, N. Venkataramam, C. M. Smith, and M. T. Gallagher, “Photonic crystal fibers,” in 27th European Conference on Optical Comunication (IEEE, 2001), paper Th. A. 2.2.

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M. Wrage, P. Glas, M. Leitner, T. Sandrock, N. N. Elkin, A. P. Napartovich, and A. G. Sukharev, “Experimental and numerical determination of coupling constant in a multicore fiber,” Opt. Commun. 175, 97-102 (2000).
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M. A. van Eijkelenborg, A. Argyros, G. Barton, I. M. Bassett, M. Fellew, G. Henry, N. A. Issa, M. C. J. Large, S. Manos, W. Padden, L. Poladian, and J. Zagari, “Recent progress in microstructured polymer optical fibre fabrication and characterization,” Opt. Fiber Technol. 9, 199-209 (2003).
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Electron. Lett. (2)

D. M. Taylor, C. R. Bennett, T. J. Shepherd, L. F. Michaille, M. D. Nielsen, and H. R. Simonsen, “Demonstration of multi-core photonic crystal fibre in an optical interconnect,” Electron. Lett. 42, 331-332 (2006).
[Crossref]

B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, and A. H. Greenaway, “Experimental study of dual-core photonic crystal fibre,” Electron. Lett. 36, 1358-1359 (2000).
[Crossref]

IEEE Photon. Technol. Lett. (2)

A. I. Siahlo, L. K. Oxenlowe, K. S. Berg, A. T. Clausen, P. A. Andersen, C. Peucheret, A. Tersigni, P. Jeppesen, K. P. Hansen, and J. R. Folkenberg, “A high-speed demultiplexer based on a nonlinear optical loop mirror with a photonic crystal fiber,” IEEE Photon. Technol. Lett. 15, 1147-1149 (2003).
[Crossref]

L. P. Shen, W.-P. Huang, G. X. Chen, and S. S. Jian, “Design and optimization of photonic crystal fibers for broad-band dispersion compensation,” IEEE Photon. Technol. Lett. 15, 540-542 (2003).
[Crossref]

J. Lightwave Technol. (3)

J. Lumin. (1)

P. St. J. Russell, “Photonic-crystal fibers,” J. Lumin. 24, 4729-4749 (2006).

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

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

Opt. Commun. (2)

M. Wrage, P. Glas, M. Leitner, T. Sandrock, N. N. Elkin, A. P. Napartovich, and A. G. Sukharev, “Experimental and numerical determination of coupling constant in a multicore fiber,” Opt. Commun. 175, 97-102 (2000).
[Crossref]

W. N. MacPherson, M. J. Gander, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, B. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Remotely addressed optical fibre curvature sensor using multicore photonic crystal fibre,” Opt. Commun. 193, 97-104 (2001).
[Crossref]

Opt. Express (8)

W. J. Wadsworth, R. M. Percival, G. Bouwmans, J. C. Knight, and P. St. J. Russell, “High power air-clad photonic crystal fibre laser,” Opt. Express 11, 48-53 (2003).
[Crossref] [PubMed]

Y. Huo, P. K. Cheo, and G. G. King, “Fundamental mode operation of a 19-core phase-locked Yb-doped fiber amplifier,” Opt. Express 12, 6230-6239 (2004).
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G. K. L. Wong, A. Y. H. Chen, S. W. Ha, R. J. Kruhlak, S. G. Murdoch, R. Leonhardt, J. D. Harvey, and N. Joly, “Characterization of chromatic dispersion in photonic crystal fibers using scalar modulation instability,” Opt. Express 13, 8662-8670 (2005).
[Crossref] [PubMed]

J. G. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth and P. St. J. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13, 534-544 (2005).
[Crossref] [PubMed]

K. Saitoh, N. J. Florous, M. Koshiba, and M. Skorobogatiy, “Design of narrow band-pass filters based on the resonant-tunneling phenomenon in multicore photonic crystal fibers,” Opt. Express 13, 10327-10335 (2005).
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F. Fogli, L. Saccomandi, P. Bassi, G. Bellanca, and S. Trillo, “Full vectorial BPM modeling of index-guiding photonic crystal fibers and couplers,” Opt. Express 10, 54-59 (2002).
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K. Saitoh, Y. Sato, and M. Koshiba, “Coupling characteristics of dual-core photonic crystal fiber couplers,” Opt. Express 11, 3188-3195 (2003).
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S. K. Varshney, N. J. Florous, K. Saitoh, and M. Koshiba, “The impact of elliptical deformations for optimizing the performance of dual-core fluorine-doped photonic crystal fiber couplers,” Opt. Express 14, 1982-1995 (2006).
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Figures (11)

Fig. 1
Fig. 1

(a) Fiber geometry; cores are marked with large circles for illustration purpose only. (b) A microscope image of the cross section of the fiber used in the experiments. (c) The cross section profile. κ 1 and κ 2 are the coupling coefficients between neighboring cores.

Fig. 2
Fig. 2

Initial amplitudes in three cores of the eigenmodes. κ 1 and κ 2 are equal.

Fig. 3
Fig. 3

(a) Amplitudes in the three cores for the decoupling mode. (b) Amplitudes in the three cores for the decoupling mode with small perturbations.

Fig. 4
Fig. 4

Decoupling versus coupling.

Fig. 5
Fig. 5

Amplitudes in the three cores versus the propagation length, with the initial light in core 2. (a) κ 1 κ 2 = 1 (b) and κ 1 κ 2 = 3 4

Fig. 6
Fig. 6

At z = 672.54 mm , the light in the central core is entirely transferred to the two side cores.

Fig. 7
Fig. 7

Amplitudes in the three cores versus the propagation length with the initial light in one side core. Light launched into (a) core 1 and (b) core 3.

Fig. 8
Fig. 8

Offset dies out at 5 cm through a simulation program.

Fig. 9
Fig. 9

A schematic of the experimental setup.

Fig. 10
Fig. 10

Normalized transmitted power through cores versus offsets along the x- and y-axes.

Fig. 11
Fig. 11

Basic result of the asymmetric coupling between cores. The fiber was 25.2 cm long.

Tables (1)

Tables Icon

Table 1 Coupling Length and the Coupling Coefficients Obtained from the Estimation, the Simulation, and the Experiments

Equations (26)

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A 1 z + 1 v g A 1 t + i β 2 2 2 A 1 t 2 = i κ 1 A 2 ,
A 2 z + 1 v g A 2 t + i β 2 2 2 A 2 t 2 = i κ 1 A 1 + i κ 2 A 3 ,
A 3 z + 1 v g A 3 t + i β 2 2 2 A 3 t 2 = i κ 2 A 2 ,
d A 1 d z = i κ 1 A 2 , d A 2 d z = i κ 1 A 1 + i κ 2 A 3 , d A 3 d z = i κ 2 A 2 .
A i ( z ) = C i 1 e i κ 1 2 + κ 2 2 z + C i 2 e i κ 1 2 + κ 2 2 z + C i 3 ,
A 1 ( z ) = κ 1 C + e i κ 1 2 + κ 2 2 z + κ 1 C e i κ 1 2 + κ 2 2 z + κ 2 C 0 ,
A 2 ( z ) = κ 1 2 + κ 2 2 C + e i κ 1 2 + κ 2 2 z κ 1 2 + κ 2 2 C e i κ 1 2 + κ 2 2 z ,
A 3 ( z ) = κ 2 C + e i κ 1 2 + κ 2 2 z + κ 2 C e i κ 1 2 + κ 2 2 z κ 1 C 0 .
C 0 = κ 2 κ 1 2 + κ 2 2 A 10 κ 1 κ 1 2 + κ 2 2 A 30 ,
C + = 1 2 ( κ 1 2 + κ 2 2 ) ( κ 1 A 10 + κ 2 A 30 ) + A 20 2 κ 1 2 + κ 2 2 ,
C = 1 2 ( κ 1 2 + κ 2 2 ) ( κ 1 A 10 + κ 2 A 30 ) A 20 2 κ 1 2 + κ 2 2 .
T = [ κ 1 2 cos ( κ 1 2 + κ 2 2 z ) + κ 2 2 κ 1 2 + κ 2 2 i κ 1 κ 1 2 + κ 2 2 sin ( κ 1 2 + κ 2 2 z ) κ 1 κ 2 κ 1 2 + κ 2 2 [ cos ( κ 1 2 + κ 2 2 z ) 1 ] i κ 1 κ 1 2 + κ 2 2 sin ( κ 1 2 + κ 2 2 z ) cos ( κ 1 2 + κ 2 2 z ) i κ 2 κ 1 2 + κ 2 2 sin ( κ 1 2 + κ 2 2 z ) κ 1 κ 2 κ 1 2 + κ 2 2 [ cos ( κ 1 2 + κ 2 2 z ) 1 ] i κ 2 κ 1 2 + κ 2 2 sin ( κ 1 2 + κ 2 2 z ) κ 2 2 cos ( κ 1 2 + κ 2 2 z ) + κ 1 2 κ 1 2 + κ 2 2 ] .
( κ 2 0 κ 1 ) , ( κ 1 κ 1 2 + κ 2 2 κ 2 ) , ( κ 1 κ 1 2 + κ 2 2 κ 2 ) .
A d A 0 κ 1 2 + κ 2 2 ( κ 2 0 κ 1 ) ,
A s A 0 2 e i κ 1 2 + κ 2 2 z κ 1 2 + κ 2 2 ( κ 1 κ 1 2 + κ 2 2 κ 2 ) ,
A a A 0 2 e i κ 1 2 + κ 2 2 z κ 1 2 + κ 2 2 ( κ 1 κ 1 2 + κ 2 2 κ 2 ) ,
A 1 ( z ) = κ 2 A 0 κ 1 2 Δ A κ 1 2 + κ 2 2 cos ( κ 1 2 + κ 2 2 z ) κ 2 2 Δ A κ 1 2 + κ 2 2 ,
A 2 ( z ) = i κ 1 Δ A κ 1 2 + κ 2 2 sin ( κ 1 2 + κ 2 2 z ) ,
A 3 ( z ) = κ 1 A 0 κ 1 κ 2 Δ A κ 1 2 + κ 2 2 cos ( κ 1 2 + κ 2 2 z ) + κ 1 κ 2 Δ A κ 1 2 + κ 2 2 .
A 1 ( z ) A 3 ( z ) = P 1 P 3 = κ 1 κ 2 ,
( A 10 A 20 A 30 ) = ( 0 1 0 ) ,
C 0 = 0 , C + = C = A 20 2 κ 1 2 + κ 2 2 .
A 1 ( z ) = i κ 1 A 20 κ 1 2 + κ 2 2 sin ( κ 1 2 + κ 2 2 z ) ,
A 2 ( z ) = A 20 cos ( κ 1 2 + κ 2 2 z ) ,
A 3 ( z ) = i κ 2 A 20 κ 1 2 + κ 2 2 sin ( κ 1 2 + κ 2 2 z ) .
κ = π V 2 k 0 n 0 a 2 exp [ ( c 0 + c 1 d ¯ + c 2 d ¯ 2 ) ] ,

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