Abstract

In this paper, detailed properties of bent solid-core photonic bandgap fibers (SC-PBGFs) are investigated. We propose an approximate equivalent straight waveguide (ESW) formulation for photonic bandgap (PBG) edges, which is convenient to see qualitatively which radiation (centripetal or centrifugal radiation) mainly occurs and the impact of bend losses for an operating wavelength. In particular, we show that cladding modes induced by bending cause several complete or incomplete leaky mode couplings with the core mode and the resultant loss peaks. Moreover, we show that the field distributions of the cladding modes are characterized by three distinct types for blue-edge, mid-gap, and red-edge wavelengths in the PBG, which is explained by considering the cladding Bloch states or resonant conditions without bending. Next, we investigate the structural dependence of the bend losses. In particular, we demonstrate the bend-loss dependence on the number of the cladding rings. Finally, by investigating the impacts of the order of PBG and the core structure on the bend losses, we discuss a tight-bending structure.

© 2009 Optical Society of America

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2008

2007

2006

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, 6059-6063 (2006).
[CrossRef]

T. Murao, K. Saitoh, and M. Koshiba, "Design of air-guiding modified honeycomb photonic band-gap fibers for effectively single-mode operation," Opt. Express 14, 2404-2412 (2006).
[CrossRef] [PubMed]

R. Amezcua-Correa, N. G. R. Broderick, M. N. Petrovich, F. Poletti, and D. J. Richardson, "Optimizing the usable bandwidth and loss through core design in realistic hollow-core photonic bandgap fibers," Opt. Express 14, 7974-7985 (2006).
[CrossRef] [PubMed]

F. Benabid, "Hollow-core photonic bandgap fibre: new light guidance for new science and technology," Phil. Trans. R. Soc. London 364, 3439-3462 (2006).
[CrossRef] [PubMed]

K. Kakihara, N. Kono, K. Saitoh, and M. Koshiba, "Full-vectorial finite element method in a cylindrical coordinate system for loss analysis of photonic wire bends," Opt. Express 14, 11128-11141 (2006).
[CrossRef] [PubMed]

T. A. Birks, G. J. Pearce, and D. M. Bird, "Approximate band structure calculation for photonic bandgap fibres," Opt. Express 14, 9483-9490 (2006).
[CrossRef] [PubMed]

A. Wang, A. K. George, and J. C. Knight, "Three-level neodymium fiber laser incorporating photonic bandgap fiber," Opt. Lett. 31, 1388-1390 (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, 5688-5698 (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, 6291-6296 (2006).
[CrossRef] [PubMed]

2005

2004

2003

1999

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance of light in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

1986

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

Alkeskjold, T. T.

Allan, D. C.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance of light in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Amezcua-Correa, R.

Anwatti, D. S.

Argyros, A.

Benabid, F.

F. Couny, F. Benabid, P. J. Roberts, M. T. Burnett, and S. A. Maier, "Identification of Bloch-modes in hollow-core photonic crystal fiber cladding," Opt. Express 15, 325-338 (2007).
[CrossRef] [PubMed]

F. Benabid, "Hollow-core photonic bandgap fibre: new light guidance for new science and technology," Phil. Trans. R. Soc. London 364, 3439-3462 (2006).
[CrossRef] [PubMed]

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 9, 858-861 (2007).
[CrossRef]

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, 6059-6063 (2006).
[CrossRef]

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, 6291-6296 (2006).
[CrossRef] [PubMed]

T. A. Birks, G. J. Pearce, and D. M. Bird, "Approximate band structure calculation for photonic bandgap fibres," Opt. Express 14, 9483-9490 (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, 5688-5698 (2006).
[CrossRef] [PubMed]

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, 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, 309-314 (2005).
[CrossRef] [PubMed]

T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. St. J. Russell, "Scaling laws and vector effects in bandgap-guiding fibres," Opt. Express 12, 69-74 (2004).
[CrossRef] [PubMed]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance of light in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Bjarklev, A.

Bouwmans, G.

Broderick, N. G. R.

Broeng, J.

Burnett, M. T.

Cordeiro, C. M. B.

Couny, F.

Cregan, R. F.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance of light in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

de Sterke, C. M.

Digonnet, M. J. F.

Douay, M.

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, 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, 819-821 (2005).
[CrossRef]

Falk, C. I.

Fan, S.

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, 819-821 (2005).
[CrossRef]

Hedley, T. D.

Hermann, D. S.

Kakihara, K.

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, 13-15 (1986).
[CrossRef]

Kokubun, Y.

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

Kono, N.

Koshiba, M.

Kuhlmey, B. T.

Lægsgaard, J.

Larsen, T. T.

Leon-Saval, S. G.

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 9, 858-861 (2007).
[CrossRef]

Litchinitser, N. M.

Lopez, F.

Luan, F.

Maier, S. A.

Mangan, B. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance of light in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

McPhedran, R. C.

Murao, T.

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, 819-821 (2005).
[CrossRef]

Olausson, C. B.

Olszewski, J.

Pearce, G. J.

Perrin, M.

Petrovich, M. N.

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, 13-15 (1986).
[CrossRef]

Poletti, F.

Pottage, J. M.

Provino, L.

Pureur, V.

Quiquempois, Y.

Richardson, D. J.

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, 819-821 (2005).
[CrossRef]

Roberts, P. J.

F. Couny, F. Benabid, P. J. Roberts, M. T. Burnett, and S. A. Maier, "Identification of Bloch-modes in hollow-core photonic crystal fiber cladding," Opt. Express 15, 325-338 (2007).
[CrossRef] [PubMed]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance of light in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Rowland, K. J.

Russell, P. S. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance of light in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Russell, P. St. J.

Saitoh, K.

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, 819-821 (2005).
[CrossRef]

Steel, M. J.

Steinvurzel, P.

Steinvurzel, P. E.

Stone, J. M.

Szpulak, M.

Urbanczyk, W.

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 9, 858-861 (2007).
[CrossRef]

White, T. P.

Appl. Phys. Lett.

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

IEEE Photon. Technol. Lett.

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, 819-821 (2005).
[CrossRef]

J. Lightwave Technol.

J. Opt. A

Y. Li, C. Wang, T. A. Birks, and D. M. Bird, "Effective index method for all-solid photonic bandgap fibres," J. Opt. A 9, 858-861 (2007).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

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]

Jpn. J. Appl. Phys.

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, 6059-6063 (2006).
[CrossRef]

Nature

J. C. Knight, "Photonic crystal fibres," Nature 424, 847-851 (2003).
[CrossRef] [PubMed]

Opt. Express

T. Murao, K. Saitoh, and M. Koshiba, "Design of air-guiding modified honeycomb photonic band-gap fibers for effectively single-mode operation," Opt. Express 14, 2404-2412 (2006).
[CrossRef] [PubMed]

R. Amezcua-Correa, N. G. R. Broderick, M. N. Petrovich, F. Poletti, and D. J. Richardson, "Optimizing the usable bandwidth and loss through core design in realistic hollow-core photonic bandgap fibers," Opt. Express 14, 7974-7985 (2006).
[CrossRef] [PubMed]

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, 1243-1251 (2003).
[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, 17935-17951 (2008).
[CrossRef] [PubMed]

P. Steinvurzel, B. T. Kuhlmey, T. P. White, M. J. Steel, C. M. de Sterke, and B. J. Eggleton, "Long wavelength anti-resonant guidance in high index inclusion microstructured fibers," Opt. Express 12, 5424-5433 (2004).
[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, 309-314 (2005).
[CrossRef] [PubMed]

T. T. Larsen, A. Bjarklev, D. S. Hermann, and J. Broeng, "Optical devices based on liquid crystal photonic bandgap fibres," Opt. Express 11, 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, 1540-1550 (2004).
[CrossRef] [PubMed]

T. T. Alkeskjold, J. Lægsgaard, A. Bjarklev, D. S. Hermann, Anwatti, J. Broeng, J. Li, and S. Wu, "All-optical modulation in dye-doped nematic liquid crystal photonic bandgap fibers," Opt. Express 12, 5857-5871 (2004).
[CrossRef] [PubMed]

T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. St. J. Russell, "Scaling laws and vector effects in bandgap-guiding fibres," Opt. Express 12, 69-74 (2004).
[CrossRef] [PubMed]

F. Couny, F. Benabid, P. J. Roberts, M. T. Burnett, and S. A. Maier, "Identification of Bloch-modes in hollow-core photonic crystal fiber cladding," Opt. Express 15, 325-338 (2007).
[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, 6291-6296 (2006).
[CrossRef] [PubMed]

A. Bétourné, V. Pureur, G. Bouwmans, Y. Quiquempois, L. Bigot, M. Perrin, and M. Douay, "Solid photonic bandgap fiber assisted by an extra air-clad structure for low-loss operation around 1.5 ?m," Opt. Express 15, 316-324 (2007).
[CrossRef] [PubMed]

K. Kakihara, N. Kono, K. Saitoh, and M. Koshiba, "Full-vectorial finite element method in a cylindrical coordinate system for loss analysis of photonic wire bends," Opt. Express 14, 11128-11141 (2006).
[CrossRef] [PubMed]

M. Perrin, Y. Quiquempois, G. Bouwmans, and M. Douay, "Coexistence of total internal reflection and bandgap modes in solid core photonic bandgap fibre with interstitial air holes," Opt. Express 15, 13783-13795 (2007).
[CrossRef] [PubMed]

T. A. Birks, G. J. Pearce, and D. M. Bird, "Approximate band structure calculation for photonic bandgap fibres," Opt. Express 14, 9483-9490 (2006).
[CrossRef] [PubMed]

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, 8452-8459 (2005).
[CrossRef] [PubMed]

C. B. Olausson, C. I. Falk, J. K. Lyngs?, 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, 13657-13662 (2008).
[CrossRef] [PubMed]

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, 2503-2511 (2005).
[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, 5688-5698 (2006).
[CrossRef] [PubMed]

J. Olszewski, M. Szpulak, and W. Urba?czyk, "Effect of coupling between fundamental and cladding modes on bending losses in photonic crystal fibers," Opt. Express 13, 6015-6022 (2005).
[CrossRef] [PubMed]

Z. Zhang Y. Shi, B. Bian, and J. Lu, "Dependence of leaky mode coupling on loss in photonic crystal fiber with hybrid cladding," Opt. Express 16, 1915-1922 (2008).
[CrossRef] [PubMed]

Opt. Lett.

Phil. Trans. R. Soc. London

F. Benabid, "Hollow-core photonic bandgap fibre: new light guidance for new science and technology," Phil. Trans. R. Soc. London 364, 3439-3462 (2006).
[CrossRef] [PubMed]

Science

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance of light in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Other

N. W. Ashcroft and N. D. Mermin, Solid State Physics, (Holt, Rinehart, and Winston, 1976).

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).

V. Pureur, A. Bétourné, G. Bouwmans, L. Bigot, M. Douay, and Y. Quiquempois, "Bending Losses in all solid 2D photonic band-gap fibers: a limiting factor?," in proceedings of 2006 European Conference on Optical Communication (ECOC), We.P3.34 (2006).

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Figures (14)

Fig. 1.
Fig. 1.

(a) Cross section of SC-PBGF, where d stands for diameter of high-index rods, Λ for distance between adjacent rods, n high for refractive index of high-index rods, and n low for that of low-index background. Inset shows schematic equivalent index profile by approximate ESW formulation with bending of the fiber. (b) Dispersion curves of the fundamental mode of this fiber for the 1st to 3rd PBGs and (c) for the 3rd PBG with enlarged illustration, respectively, where the green color corresponds to the PBG region and the black dashed line to the silica-core index. (d) Confinement losses as a function of wavelength for the SC-PBGF with d/Λ = 0.4, Λ = 6.75 μm, n high = 1.48, n low = 1.45, and 6 cladding rings, respectively.

Fig. 2.
Fig. 2.

Equivalent PBG edges as a function of x at the bend radius corresponding to the first appearance of the complete mode coupling (discussed in Fig. 3) for (a) blue-edge and (b) red-edge wavelengths, respectively.

Fig. 3.
Fig. 3.

Bend-radius dependence of the effective refractive indices of the modes for the (a) blue-edge, (b) mid-gap, and (c) red-edge wavelengths, respectively.

Fig. 4.
Fig. 4.

x-component of electric field distributions of horizontally polarized modes at the bend radius corresponding to the first appearance of the complete mode coupling (in Fig. 3) for the (a) blue-edge, (b) mid-gap, and (c) red-edge wavelengths, respectively. The rod modes consist of LP12 mode for blue-edge and mid-gap, and LP02 mode for red-edge wavelength, respectively.

Fig. 5.
Fig. 5.

Bend losses as a function of bend radius for (a) blue-edge and red-edge wavelengths, and for (b) mid-gap and red-edge wavelengths, respectively.

Fig. 6.
Fig. 6.

x-component of the electric field distributions of the horizontally polarized modes at the loss peaks (in Fig. 5(b)) due to mode coupling between core and cladding modes for mid-gap wavelength.

Fig. 7.
Fig. 7.

Bend-radius dependence of the effective refractive indices of the modes for the fiber with 5 cladding rings for the (a) blue-edge, (b) mid-gap, and (c) red-edge wavelengths, respectively.

Fig. 8.
Fig. 8.

Bend losses as a function of bend radius for the fiber with 5 cladding rings for (a) blue-edge and red-edge wavelengths, and for (b) mid-gap and red-edge wavelengths, respectively.

Fig. 9.
Fig. 9.

Equivalent PBG edges as a function of x at the bend radius corresponding to the first appearance of the complete mode coupling (in Fig. 7(c)) for the red-edge wavelength.

Fig. 10.
Fig. 10.

x-component of the electric field distributions of the horizontally polarized modes at the loss peaks (in Fig. 8(b)) due to mode coupling between core and cladding modes for mid-gap wavelength for the 5-cladding-rings SC-PBGF with d/Λ = 0.4 and Λ = 6.75 μm (same as Fig. 6 except for the number of the cladding rings).

Fig. 11.
Fig. 11.

Bend losses as a function of bend radius for the fiber with 8 cladding rings for (a) blue-edge and (b) mid-gap wavelengths. The bend losses for 6 cladding rings (shown in Fig. 5) are also shown with dotted curves as a reference.

Fig. 12.
Fig. 12.

Comparison of bend losses as a function of bend radius for the mid-gap wavelength of 1st, 2nd and 3rd PBGs for the structures with (a) 1 cell core and 6 cladding rings, and (b) 7 cell core and 5 cladding rings (cladding region is in keeping with that of the fiber with 1 cell core and 6 cladding rings).

Fig. 13.
Fig. 13.

Bend losses as a function of bend radius for the fibers with d/Λ = 0.4, Λ = 6.75 μm, 7 cell core and 5 cladding rings (solid black curve), and with d/Λ = 0.56, Λ = 13.5 μm, 1 cell core and 6 cladding rings (dashed blue curve) at λ = 1.5 μm (mid-gap wavelength of the fibers). The core radius relative to wavelength is almost identical for the fibers. As a reference, bend losses for the fibers with d/Λ = 0.4, Λ = 6.75 μm, 1 cell core and 6 cladding rings (dashed black curve), and with d/Λ = 0.56, Λ = 13.5 μm, 7 cell core and 5 cladding rings (solid blue curve) are also depicted at the same time.

Fig. 14.
Fig. 14.

Comparison of bend losses as a function of bend radius for the d/Λ = 0.4, n high = 1.48 and d/Λ = 0.35, n high = 1.49 at the mid-gap wavelength of (a) 3rd PBG for 1 cell-core structure, (b) 1st PBG for 1 cell-core structure, and (c) 1st PBG for 7 cell-core structure. The pitch for both structures is Λ = 6.75 μm.

Tables (1)

Tables Icon

Table 1. Comparison of results for gradient [1/μm] of PBG edges between FEM (in Fig. 2) and approximation (n edge/R).

Equations (5)

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n eq ( x , y ) = n ( x , y ) ( 1 + x / R ) ,
V eq = ( 1 + x / R ) V ,
V = πd n high 2 n low 2 / λ ,
Δ eq ( x , y ) = n high eq 2 n low eq 2 2 n high eq 2 = n high 2 n low 2 2 n high 2 = Δ ,
n edge eq ( x ) = n edge ( 1 + x / R ) .

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