Abstract

In this paper, we propose a novel mechanism for suppression of higher-order modes (HOMs), namely multiple resonant coupling, in all-solid photonic bandgap fibers (PBGFs) with effectively large core diameters. In an analogy to the well-known tight-binding theory in solid-state physics, multiple anti-resonant reflecting optical waveguide (ARROW) modes bound in designedly arranged defects in the cladding make up Bloch states and resultant photonic bands with a finite effective-index width, which contribute to the suppression of HOMs. In particular, contrary to the conventional method for the HOM suppression using the index-matching of the HOMs in the core of the PBGF and the defect mode arranged in the cladding, the proposed mechanism guarantees a broadband HOM suppression without a precise structural design. This effect is explained by the multiple resonant coupling, as well as an enhanced confinement loss mechanism which occurs near the condition satisfying the multiple resonant coupling. Moreover, we show that the proposed structure exhibits a lower bending loss characteristic when compared to the conventional all-solid PBGFs. The simultaneous realization of the single-mode operation and the low bending loss property is due to the novel cladding concept named as heterostructured cladding. The proposed structure also resolves the issue for the increased confinement loss property in the first-order photonic bandgap (PBG) at the same time.

©2011 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  39. Y. Li, C. Wang, T. A. Birks, and D. M. Bird, “Effective index method for all-solid photonic bandgap fibres,” J. Opt. A: Pure Appl. Opt. 9, 858–861 (2007).
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2010 (1)

2009 (2)

2008 (3)

2007 (6)

2006 (7)

2005 (5)

2004 (4)

2003 (2)

2002 (1)

K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: Application to photonic crystal fibers,” IEEE J. Quantum Electron. 38(7), 927–933 (2002).
[Crossref]

1989 (1)

J. D. Love, “Application of a low-loss criterion to optical waveguides and devices,” IEE Proc. Pt. J 136, 225–228 (1989).

1988 (1)

T. Baba, Y. Kokubun, T. Sakaki, and K. Iga, “Loss reduction of an ARROW waveguide in shorter wavelength and its stack configuration,” J. Lightwave Technol. 6(9), 1440–1445 (1988).
[Crossref]

1986 (1)

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49(1), 13–15 (1986).
[Crossref]

Afshar V, S.

Alkeskjold, T. T.

Anawati, A.

Argyros, A.

Baba, T.

T. Baba, Y. Kokubun, T. Sakaki, and K. Iga, “Loss reduction of an ARROW waveguide in shorter wavelength and its stack configuration,” J. Lightwave Technol. 6(9), 1440–1445 (1988).
[Crossref]

Bétourné, A.

Bigot, L.

Bird, D. M.

Birks, T. A.

Y. Li, C. Wang, T. A. Birks, and D. M. Bird, “Effective index method for all-solid photonic bandgap fibres,” J. Opt. A: Pure Appl. Opt. 9, 858–861 (2007).
[Crossref]

Z. Wang, T. Taru, T. A. Birks, J. C. Knight, Y. Liu, and J. Du, “Coupling in dual-core photonic bandgap fibers: theory and experiment,” Opt. Express 15(8), 4795–4803 (2007).
[Crossref] [PubMed]

T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express 14(20), 9483–9490 (2006).
[Crossref] [PubMed]

J. M. Stone, G. J. Pearce, F. Luan, T. A. Birks, J. C. Knight, A. K. George, and D. M. Bird, “An improved photonic bandgap fiber based on an array of rings,” Opt. Express 14(13), 6291–6296 (2006).
[Crossref] [PubMed]

T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, and D. M. Bird, “Bend loss in all-solid bandgap fibres,” Opt. Express 14(12), 5688–5698 (2006).
[Crossref] [PubMed]

J. C. Knight, F. Luan, G. J. Pearce, A. Wang, T. A. Birks, and D. M. Bird, “Solid photonic bandgap fibres and applications,” Jpn. J. Appl. Phys. 45(No. 8A), 6059–6063 (2006).
[Crossref]

A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, and P. St J Russell, “Guidance properties of low-contrast photonic bandgap fibres,” Opt. Express 13(7), 2503–2511 (2005).
[Crossref] [PubMed]

A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, and P. St. J. Russell, “Photonic bandgap with an index step of one percent,” Opt. Express 13(1), 309–314 (2005).
[Crossref] [PubMed]

Bjarklev, A.

Bouwmans, G.

Broeng, J.

Cordeiro, C. M. B.

de Sterke, C. M.

Digonnet, M. J. F.

Douay, M.

Du, J.

Duguay, M. A.

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49(1), 13–15 (1986).
[Crossref]

Dunn, S. C.

Eggleton, B. J.

Engan, H. E.

M. W. Haakestad, T. T. Alkeskjold, M. D. Nielsen, L. Scolari, J. Riishede, H. E. Engan, and A. Bjarklev, “Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber,” IEEE Photon. Technol. Lett. 17(4), 819–821 (2005).
[Crossref]

Falk, C. I.

Fan, S.

Fini, J. M.

Florous, N. J.

George, A. K.

Haakestad, M. W.

M. W. Haakestad, T. T. Alkeskjold, M. D. Nielsen, L. Scolari, J. Riishede, H. E. Engan, and A. Bjarklev, “Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber,” IEEE Photon. Technol. Lett. 17(4), 819–821 (2005).
[Crossref]

Hansen, K. P.

Hedley, T. D.

Hermann, D. S.

Iga, K.

T. Baba, Y. Kokubun, T. Sakaki, and K. Iga, “Loss reduction of an ARROW waveguide in shorter wavelength and its stack configuration,” J. Lightwave Technol. 6(9), 1440–1445 (1988).
[Crossref]

Jaouen, Y.

V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett. 92(6), 061113 (2008).
[Crossref]

Jensen, B. B.

Kim, H. K.

Kino, G. S.

Knight, J. C.

Koch, T. L.

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49(1), 13–15 (1986).
[Crossref]

Kokubun, Y.

T. Baba, Y. Kokubun, T. Sakaki, and K. Iga, “Loss reduction of an ARROW waveguide in shorter wavelength and its stack configuration,” J. Lightwave Technol. 6(9), 1440–1445 (1988).
[Crossref]

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49(1), 13–15 (1986).
[Crossref]

Koshiba, M.

T. Murao, K. Saitoh, and M. Koshiba, “Detailed theoretical investigation of bending properties in solid-core photonic bandgap fibers,” Opt. Express 17(9), 7615–7629 (2009).
[Crossref] [PubMed]

K. Saitoh, N. J. Florous, T. Murao, and M. Koshiba, “Design of photonic band gap fibers with suppressed higher-order modes: towards the development of effectively single mode large hollow-core fiber platforms,” Opt. Express 14(16), 7342–7352 (2006).
[Crossref] [PubMed]

K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: Application to photonic crystal fibers,” IEEE J. Quantum Electron. 38(7), 927–933 (2002).
[Crossref]

T. Murao, K. Nagao, K. Saitoh, and M. Koshiba, “Understanding formation of photonic bandgap edge for maximum propagation angle in all-solid photonic bandgap fibers,” J. Opt. Soc. Am. B (accepted for publication).

Kuhlmey, B. T.

Lægsgaard, J.

Larsen, T. T.

Leon-Saval, S. G.

Li, J.

Li, Y.

Y. Li, C. Wang, T. A. Birks, and D. M. Bird, “Effective index method for all-solid photonic bandgap fibres,” J. Opt. A: Pure Appl. Opt. 9, 858–861 (2007).
[Crossref]

Litchinitser, N. M.

Liu, Y.

Lopez, F.

Love, J. D.

J. D. Love, “Application of a low-loss criterion to optical waveguides and devices,” IEE Proc. Pt. J 136, 225–228 (1989).

Luan, F.

Luo, J.

Lyngsø, J. K.

Maruyama, H.

McPhedran, R. C.

Monro, T. M.

Murao, T.

Nagao, K.

T. Murao, K. Nagao, K. Saitoh, and M. Koshiba, “Understanding formation of photonic bandgap edge for maximum propagation angle in all-solid photonic bandgap fibers,” J. Opt. Soc. Am. B (accepted for publication).

Nielsen, M. D.

M. W. Haakestad, T. T. Alkeskjold, M. D. Nielsen, L. Scolari, J. Riishede, H. E. Engan, and A. Bjarklev, “Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber,” IEEE Photon. Technol. Lett. 17(4), 819–821 (2005).
[Crossref]

Olausson, C. B.

Pearce, G. J.

Perrin, M.

Pfeiffer, L.

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49(1), 13–15 (1986).
[Crossref]

Provino, L.

Pureur, V.

Quiquempois, Y.

Ren, G.

Riishede, J.

M. W. Haakestad, T. T. Alkeskjold, M. D. Nielsen, L. Scolari, J. Riishede, H. E. Engan, and A. Bjarklev, “Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber,” IEEE Photon. Technol. Lett. 17(4), 819–821 (2005).
[Crossref]

Rowland, K. J.

Russell, P. St. J.

Saitoh, K.

T. Murao, K. Saitoh, and M. Koshiba, “Detailed theoretical investigation of bending properties in solid-core photonic bandgap fibers,” Opt. Express 17(9), 7615–7629 (2009).
[Crossref] [PubMed]

K. Saitoh, N. J. Florous, T. Murao, and M. Koshiba, “Design of photonic band gap fibers with suppressed higher-order modes: towards the development of effectively single mode large hollow-core fiber platforms,” Opt. Express 14(16), 7342–7352 (2006).
[Crossref] [PubMed]

K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: Application to photonic crystal fibers,” IEEE J. Quantum Electron. 38(7), 927–933 (2002).
[Crossref]

T. Murao, K. Nagao, K. Saitoh, and M. Koshiba, “Understanding formation of photonic bandgap edge for maximum propagation angle in all-solid photonic bandgap fibers,” J. Opt. Soc. Am. B (accepted for publication).

Sakaki, T.

T. Baba, Y. Kokubun, T. Sakaki, and K. Iga, “Loss reduction of an ARROW waveguide in shorter wavelength and its stack configuration,” J. Lightwave Technol. 6(9), 1440–1445 (1988).
[Crossref]

Scolari, L.

M. W. Haakestad, T. T. Alkeskjold, M. D. Nielsen, L. Scolari, J. Riishede, H. E. Engan, and A. Bjarklev, “Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber,” IEEE Photon. Technol. Lett. 17(4), 819–821 (2005).
[Crossref]

Shirakawa, A.

Shum, P.

St J Russell, P.

Steinvurzel, P. E.

Stone, J. M.

Taru, T.

Therkildsen, K. T.

Thomsen, J. W.

Tong, W.

Ueda, K.

Usner, B.

Wang, A.

Wang, C.

Y. Li, C. Wang, T. A. Birks, and D. M. Bird, “Effective index method for all-solid photonic bandgap fibres,” J. Opt. A: Pure Appl. Opt. 9, 858–861 (2007).
[Crossref]

Wang, Z.

White, T. P.

Wu, S. T.

Yu, X.

Zhang, L.

Appl. Phys. Lett. (2)

V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett. 92(6), 061113 (2008).
[Crossref]

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49(1), 13–15 (1986).
[Crossref]

IEE Proc. Pt. J (1)

J. D. Love, “Application of a low-loss criterion to optical waveguides and devices,” IEE Proc. Pt. J 136, 225–228 (1989).

IEEE J. Quantum Electron. (1)

K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: Application to photonic crystal fibers,” IEEE J. Quantum Electron. 38(7), 927–933 (2002).
[Crossref]

IEEE Photon. Technol. Lett. (1)

M. W. Haakestad, T. T. Alkeskjold, M. D. Nielsen, L. Scolari, J. Riishede, H. E. Engan, and A. Bjarklev, “Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber,” IEEE Photon. Technol. Lett. 17(4), 819–821 (2005).
[Crossref]

J. Lightwave Technol. (2)

T. Baba, Y. Kokubun, T. Sakaki, and K. Iga, “Loss reduction of an ARROW waveguide in shorter wavelength and its stack configuration,” J. Lightwave Technol. 6(9), 1440–1445 (1988).
[Crossref]

M. J. F. Digonnet, H. K. Kim, G. S. Kino, and S. Fan, “Understanding air-core photonic-bandgap fibers: Analogy to conventional fibers,” J. Lightwave Technol. 23(12), 4169–4177 (2005).
[Crossref]

J. Opt. A: Pure Appl. Opt. (2)

J. Lægsgaard, “Gap formation and guided modes in photonic bandgap fibres with high-index rods,” J. Opt. A: Pure Appl. Opt. 6, 798–804 (2004).
[Crossref]

Y. Li, C. Wang, T. A. Birks, and D. M. Bird, “Effective index method for all-solid photonic bandgap fibres,” J. Opt. A: Pure Appl. Opt. 9, 858–861 (2007).
[Crossref]

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

T. Murao, K. Nagao, K. Saitoh, and M. Koshiba, “Understanding formation of photonic bandgap edge for maximum propagation angle in all-solid photonic bandgap fibers,” J. Opt. Soc. Am. B (accepted for publication).

Jpn. J. Appl. Phys. (1)

J. C. Knight, F. Luan, G. J. Pearce, A. Wang, T. A. Birks, and D. M. Bird, “Solid photonic bandgap fibres and applications,” Jpn. J. Appl. Phys. 45(No. 8A), 6059–6063 (2006).
[Crossref]

Opt. Express (19)

N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Resonances in microstructured optical waveguides,” Opt. Express 11(10), 1243–1251 (2003).
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T. T. Larsen, A. Bjarklev, D. S. Hermann, and J. Broeng, “Optical devices based on liquid crystal photonic bandgap fibres,” Opt. Express 11(20), 2589–2596 (2003).
[Crossref] [PubMed]

N. M. Litchinitser, S. C. Dunn, P. E. Steinvurzel, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Application of an ARROW model for designing tunable photonic devices,” Opt. Express 12(8), 1540–1550 (2004).
[Crossref] [PubMed]

Z. Wang, T. Taru, T. A. Birks, J. C. Knight, Y. Liu, and J. Du, “Coupling in dual-core photonic bandgap fibers: theory and experiment,” Opt. Express 15(8), 4795–4803 (2007).
[Crossref] [PubMed]

C. B. Olausson, C. I. Falk, J. K. Lyngsø, B. B. Jensen, K. T. Therkildsen, J. W. Thomsen, K. P. Hansen, A. Bjarklev, and J. Broeng, “Amplification and ASE suppression in a polarization-maintaining ytterbium-doped all-solid photonic bandgap fibre,” Opt. Express 16(18), 13657–13662 (2008).
[Crossref] [PubMed]

K. J. Rowland, S. Afshar V, and T. M. Monro, “Bandgaps and antiresonances in integrated-ARROWs and Bragg fibers; a simple model,” Opt. Express 16(22), 17935–17951 (2008).
[Crossref] [PubMed]

A. Shirakawa, H. Maruyama, K. Ueda, C. B. Olausson, J. K. Lyngsø, and J. Broeng, “High-power Yb-doped photonic bandgap fiber amplifier at 1150-1200 nm,” Opt. Express 17(2), 447–454 (2009).
[Crossref] [PubMed]

T. Murao, K. Saitoh, and M. Koshiba, “Detailed theoretical investigation of bending properties in solid-core photonic bandgap fibers,” Opt. Express 17(9), 7615–7629 (2009).
[Crossref] [PubMed]

V. Pureur, J. C. Knight, and B. T. Kuhlmey, “Higher order guided mode propagation in solid-core photonic bandgap fibers,” Opt. Express 18(9), 8906–8915 (2010).
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T. T. Alkeskjold, J. Lægsgaard, A. Bjarklev, D. S. Hermann, A. Anawati, J. Broeng, J. Li, and S. T. Wu, “All-optical modulation in dye-doped nematic liquid crystal photonic bandgap fibers,” Opt. Express 12(24), 5857–5871 (2004).
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A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, and P. St. J. Russell, “Photonic bandgap with an index step of one percent,” Opt. Express 13(1), 309–314 (2005).
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A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, and P. St J Russell, “Guidance properties of low-contrast photonic bandgap fibres,” Opt. Express 13(7), 2503–2511 (2005).
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G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, “Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm,” Opt. Express 13(21), 8452–8459 (2005).
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T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, and D. M. Bird, “Bend loss in all-solid bandgap fibres,” Opt. Express 14(12), 5688–5698 (2006).
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J. M. Stone, G. J. Pearce, F. Luan, T. A. Birks, J. C. Knight, A. K. George, and D. M. Bird, “An improved photonic bandgap fiber based on an array of rings,” Opt. Express 14(13), 6291–6296 (2006).
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K. Saitoh, N. J. Florous, T. Murao, and M. Koshiba, “Design of photonic band gap fibers with suppressed higher-order modes: towards the development of effectively single mode large hollow-core fiber platforms,” Opt. Express 14(16), 7342–7352 (2006).
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T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express 14(20), 9483–9490 (2006).
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Other (5)

T. Murao, K. Saitoh, K. Nagao, and M. Koshiba, “Design principle for low bending losses in all-solid photonic bandgap fibers,” in Proceedings of Conference on Lasers and Electro-Optics and Quantum Electronics and Laser Science Conference (CLEO/QELS2010), paper JTuD45.

T. Murao, K. Saitoh, T. Taru, T. Nagashima, K. Maeda, T. Sasaki, and M. Koshiba, “Bend-insensitive and effectively single-moded all-solid photonic bandgap fibers with heterostructured cladding,” in Proceedings of European Conference on Optical Communication (ECOC), 2.1.4 (2009).

R. Goto, K. Takenaga, S. Matsuo, and K. Himeno, “Solid photonic band-gap fiber with 400 nm bandwidth and loss below 4 dB/km at 1520 nm,” in Proceedings of 2007 Optical Fiber Communication Conference and Exposition/National Fiber Optic Engineers Conference (OFC/NFOEC), OML7 (2007).

T. Taru, J. Hou, and J. C. Knight, “Raman gain suppression in all-solid photonic bandgap fiber,” in Proceedings of 2007 European Conference on Optical Communication (ECOC), 7.1.1 (2007).

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Figures (11)

Fig. 1
Fig. 1 Cross sections of typical all-solid PBGF (“uniform” structure) with (a) 1-cell core and 7 rings, and (b) 7-cell core with 6 rings (cladding region is in keeping with that of the fiber with 1-cell core and 7 rings), where d stands for diameter of high-index rods, Λ is the distance between adjacent rods, and n high, n low are refractive indices of high-index rods and low-index background, respectively. Unless the exceptional clause is shown, the structural parameters of d/Λ = 0.4, Λ = 7.0 μm, n high = 1.48, and n low = 1.45 are considered.

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Fig. 2
Fig. 2 Confinement losses as a function of wavelength for uniform 1-cell-core structure (red curve) and 7-cell-core structure (blue curves) in the first-order PBG. Exploiting the 7-cell-core structure in the first-order PBG also leads to the improvement of the confinement loss property as well as the bending loss property, while causing the multi-mode operation.

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Fig. 3
Fig. 3 Schematic representation of the dispersion relation for (a) the conventional HOM suppression method based on index-matching operating at only a particular wavelength that index-matching occurs, (b) the proposed method based on multiple resonant coupling operating for a wide wavelength range owing to photonic bands generated by ARROW modes in the cladding, and (c) the enhanced confinement loss mechanism which supports the HOM suppression at the condition that multiple resonant coupling does not occur.

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Fig. 4
Fig. 4 (a) Cross section for the proposed all-solid PBGF which satisfies the multiple resonant coupling shown in Fig. 3(b) or enhanced confinement loss mechanism shown in Fig. 3(c). The cladding has periodically arranged multiple cores. (b) Confinement losses as a function of wavelength for the proposed structure with 6 cladding rings. The HOM is effectively suppressed while keeping the confinement loss low for the fundamental-like mode for d/Λ = 0.4 and Λ = 7.0 μm (blue curves), as well as for d/Λ = 0.5 and Λ = 5.6 μm (red curves).

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Fig. 5
Fig. 5 Bending losses as function of bending radius at λ = 1.55 μm for the proposed structure shown in Fig. 4(a) (red curve) and modified structure shown in Fig. 6(a) (purple curve), where d/Λ = 0.5 and Λ = 5.6 μm. As a reference, results are also shown for conventional 1-cell-core structure with triangular lattice cladding utilizing the first-order PBG shown in Fig. 1(a) (blue curve), where d/Λ = 0.4 and Λ = 7.0 μm, and conventional 1-cell-core structure utilizing the third-order PBG (cyan curve). For the structure utilizing the third-order PBG, the parameters are chosen as d/Λ = 0.56 and Λ = 13.5 μm with 6 cladding rings.

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Fig. 6
Fig. 6 (a) Cross section of the modified heterostructured PBGF, where the several high-index rods are embedded in the low-index regions designed as causing the multiple resonant coupling mechanism. (b) Confinement losses as a function of wavelength for the guided modes with 6 cladding rings. The HOM is effectively suppressed while keeping the confinement loss low for the fundamental-like mode for d/Λ = 0.4 and Λ = 7.0 μm (blue curves), as well as for d/Λ = 0.5 and Λ = 5.6 μm (red curves).

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Fig. 7
Fig. 7 Photonic band diagrams (green region) for the (a) triangular lattice and (b) honeycomb lattice for first, second, and third-order PBGs (from left to right), where red curves in (a) are dispersion curves of the fundamental-like mode for the 1-cell-core all-solid PBGF shown in Fig. 1(a) and the arrows in (b) correspond to the condition at which Bloch states will be presented in Fig. 8. The insets in (a) and (b) depict the corresponding lattice structure, where the hexagonal region stands for the unit cell. (c) First Brillouin zone boundary in reciprocal lattice space for two-dimensional triangular lattice with the pitch Λ′, where (kx , ky ) = (0, 0) at Γ point, (kx , ky ) = ( 2 π / 3 Λ ' ,   0 ) at M point, and (kx , ky ) = ( 2 π / 3 Λ ' ,   2 π / 3 Λ ' ) at K point, where Λ′ = 3 Λ .

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Fig. 8
Fig. 8 One of the transverse electric field components (Ex ) of Bloch states corresponding to the photonic bands’ edge indicated by arrows from A to G in Fig. 7(b) is depicted from (a) to (g), respectively. The states of A, B, C, D, E, F, and G in Fig. 7(b) correspond to K, Γ, Γ, K, M, K, and Γ, respectively, and are composed of even state of LP11 rod modes, even state of LP02 rod modes, even state of LP02 rod modes, even state of LP12 rod modes, odd state of LP11 rod modes, even state of LP12 rod modes, and even state of LP03 rod modes, respectively.

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Fig. 9
Fig. 9 (a) Cross section of conventional all-solid PBGF coupler with 2 cores. (b) Dispersion curves of the x-polarized even (red curves) and odd modes (blue curves) for the coupler, where the region with green represents the PBG of the triangular lattice for the first, second, and third-order PBGs (from left to right). Apparently, the effective index for the odd mode is larger in the odd-ordered PBGs.

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Fig. 10
Fig. 10 Cross sections of the (a) uniform 7-cell-core structure with 2 rings which composes the 1st cladding and (b) structure with uniform honeycomb lattice cladding which composes the 2nd cladding of the heterostructure shown in Fig. 4(a).

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Fig. 11
Fig. 11 (a) Dispersion curves of the fundamental-like and the HOM (HE21 mode) for the (a) 2nd cladding fiber shown in Fig. 10(b), and (b) proposed fiber with the heterostructured cladding shown in Fig. 4(a) in the first-order PBG (red curves), where the green region stands for the PBG of honeycomb lattice.

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Equations (1)

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V = π d λ ( n h i g h 2 − n l o w 2 ) 1 2

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