Abstract

The development of highly dispersive lower and higher order cladding modes and their degeneration with respect to the fundamental core mode in a bent photonic crystal fiber is rigorously studied by use of the full-vectorial finite element method. It is shown that changes in the bending radius can modify the modal properties of large-area photonic crystal fibers, important for a number of potential practical applications.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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  36. M. Koshiba and K. Saitoh, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,” IEEE J. Quantum Electron. 38927-933 (2002).
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2008 (2)

2007 (3)

2006 (2)

X. Jiang, W. Qi, H. Zhang, Y. Tang, Y. Hao, J. Yang, and M. Wang, “Loss crosstalk 1×2 thermooptic digital optical switch with integrated S-bend attenuator,” IEEE Photon. Technol. Lett. 18, 610-612 (2006).
[CrossRef]

P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24, 4729-4749 (2006).
[CrossRef]

2005 (5)

2004 (1)

J. Arriaga, J. C. Knight, and P. St. J. Russell, “Modeling the propagation of light in photonic crystal fibers,” Physica D (Amsterdam) 189, 100-106 (2004).
[CrossRef]

2003 (3)

2002 (3)

M. Koshiba and K. Saitoh, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,” IEEE J. Quantum Electron. 38927-933 (2002).
[CrossRef]

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).
[PubMed]

P. Bienstman, E. Six, M. Roelens, M. Vanwolleghem, and R. Baets, “Calculation of bending losses in dielectric waveguides using eigenmode expansion and perfectly matched layers,” IEEE Photon. Technol. Lett. 14, 164-166(2002).
[CrossRef]

2001 (1)

2000 (3)

M. Rajarajan, S. S. A. Obayya, B. M. A. Rahman, K. T. V. Grattan, and H. A. El-Mikathi, “Characterization of low-loss waveguide bends with offset optimisation for compact photonic integrated circuits,” IEE Proc. Optoelectron. 147, 382-388 (2000).
[CrossRef]

W. Berglund and A. Gopinath, “WKB analysis of bend losses in optical waveguides,” J. Lightwave Technol. 18, 1161-1166(2000).
[CrossRef]

M. van Eijkelenborg, J. Canning, T. Ryan, and K. Lyytikainen, “Bending-induced coloring in a photonic crystal fiber,” Opt. Express 7, 88-94 (2000).
[CrossRef] [PubMed]

1999 (2)

1998 (2)

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]

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microwave Guid. Wave Lett. 8, 223-225 (1998).
[CrossRef]

1997 (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

1996 (1)

S. Kim and A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol. 14, 2085-2092 (1996).
[CrossRef]

1993 (3)

T. Yamamoto and M. Koshiba, “Numerical analysis of curvature loss in optical waveguides by the finite-element method,” J. Lightwave Technol. 11, 1579-1583 (1993).
[CrossRef]

M. Zirngibl, C. H. Joyner, L. W. Stulz, Th. Gaiffe, and C. Dragone, “Polarization independent 8×8 waveguide grating multiplexers on InP,” Electron. Lett. 29, 201-202 (1993).
[CrossRef]

R. R. Hayes and D. Yap, “GaAs spiral optical waveguides for delay-line applications,” J. Lightwave Technol. 11, 523-528(1993).
[CrossRef]

1989 (1)

J. Gu, P. Besse, and H. Melchior, “Novel method for analysis of curved optical rib-waveguides,” Electron. Lett. 25, 278-280(1989).
[CrossRef]

1987 (1)

1984 (2)

H. F. Taylor, “Bending effects in optical fibers,” J. Lightwave Technol. 2, 617-628 (1984).
[CrossRef]

B. M. A. Rahman and J. B. Davies, “Finite-element solution of integrated optical waveguides,” J. Lightwave Technol. 2, 682-688 (1984).
[CrossRef]

1983 (1)

1975 (1)

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75-83 (1975).
[CrossRef]

Argyros, A.

Arriaga, J.

J. Arriaga, J. C. Knight, and P. St. J. Russell, “Modeling the propagation of light in photonic crystal fibers,” Physica D (Amsterdam) 189, 100-106 (2004).
[CrossRef]

Baets, R.

P. Bienstman, E. Six, M. Roelens, M. Vanwolleghem, and R. Baets, “Calculation of bending losses in dielectric waveguides using eigenmode expansion and perfectly matched layers,” IEEE Photon. Technol. Lett. 14, 164-166(2002).
[CrossRef]

R. Baets and P. E. Lagasse, “Loss calculation and design of arbitrarily curved integrated-optic waveguides,” J. Opt. Soc. Am. 73, 177-182 (1983).
[CrossRef]

Bassi, P.

Bellanca, G.

Bennett, P. J.

Berghmans, F.

Berglund, W.

Besse, P.

J. Gu, P. Besse, and H. Melchior, “Novel method for analysis of curved optical rib-waveguides,” Electron. Lett. 25, 278-280(1989).
[CrossRef]

Bienstman, P.

P. Bienstman, E. Six, M. Roelens, M. Vanwolleghem, and R. Baets, “Calculation of bending losses in dielectric waveguides using eigenmode expansion and perfectly matched layers,” IEEE Photon. Technol. Lett. 14, 164-166(2002).
[CrossRef]

Birks, T.

Birks, T. A.

Botten, L. C.

Broderick, N. G. R.

Canning, J.

Chew, W. C.

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microwave Guid. Wave Lett. 8, 223-225 (1998).
[CrossRef]

Chu, S. T.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

Cordeiro, C. M. B.

Davies, J. B.

B. M. A. Rahman and J. B. Davies, “Finite-element solution of integrated optical waveguides,” J. Lightwave Technol. 2, 682-688 (1984).
[CrossRef]

de Sandro, J. P.

de Sterke, C. M.

Dragone, C.

M. Zirngibl, C. H. Joyner, L. W. Stulz, Th. Gaiffe, and C. Dragone, “Polarization independent 8×8 waveguide grating multiplexers on InP,” Electron. Lett. 29, 201-202 (1993).
[CrossRef]

El-Mikathi, H. A.

M. Rajarajan, S. S. A. Obayya, B. M. A. Rahman, K. T. V. Grattan, and H. A. El-Mikathi, “Characterization of low-loss waveguide bends with offset optimisation for compact photonic integrated circuits,” IEE Proc. Optoelectron. 147, 382-388 (2000).
[CrossRef]

Florous, N. J.

Fogli, F.

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

Gaiffe, Th.

M. Zirngibl, C. H. Joyner, L. W. Stulz, Th. Gaiffe, and C. Dragone, “Polarization independent 8×8 waveguide grating multiplexers on InP,” Electron. Lett. 29, 201-202 (1993).
[CrossRef]

Ghatak, A. K.

Golojuch, G.

Gopinath, A.

W. Berglund and A. Gopinath, “WKB analysis of bend losses in optical waveguides,” J. Lightwave Technol. 18, 1161-1166(2000).
[CrossRef]

S. Kim and A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol. 14, 2085-2092 (1996).
[CrossRef]

Grattan, K. T. V.

N. Kejalakshmy, B. M. A. Rahman, A. K. M. S. Kabir, M. Rajarajan, and K. T. V. Grattan, “Single mode operation of photonic crystal fiber using a full vectorial finite element method,” Proc. SPIE 6588, 65880T (2007).
[CrossRef]

B. M. A. Rahman, A. K. M. S. Kabir, M. Rajarajan, and K. T. V. Grattan, “Finite element modal solutions of planar photonic crystal fibers with rectangular air-holes,” Opt. Quantum Electron. 37, 171-183 (2005).
[CrossRef]

M. Rajarajan, S. S. A. Obayya, B. M. A. Rahman, K. T. V. Grattan, and H. A. El-Mikathi, “Characterization of low-loss waveguide bends with offset optimisation for compact photonic integrated circuits,” IEE Proc. Optoelectron. 147, 382-388 (2000).
[CrossRef]

Gu, J.

J. Gu, P. Besse, and H. Melchior, “Novel method for analysis of curved optical rib-waveguides,” Electron. Lett. 25, 278-280(1989).
[CrossRef]

Guobin, R.

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech House, 2000).

Hao, Y.

X. Jiang, W. Qi, H. Zhang, Y. Tang, Y. Hao, J. Yang, and M. Wang, “Loss crosstalk 1×2 thermooptic digital optical switch with integrated S-bend attenuator,” IEEE Photon. Technol. Lett. 18, 610-612 (2006).
[CrossRef]

Harris, J. H.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75-83 (1975).
[CrossRef]

Haus, H. A.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

Hayes, R. R.

R. R. Hayes and D. Yap, “GaAs spiral optical waveguides for delay-line applications,” J. Lightwave Technol. 11, 523-528(1993).
[CrossRef]

Heiblum, M.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75-83 (1975).
[CrossRef]

Hwang, I.

Iizawa, K.

Jedidi, R.

Jiang, X.

X. Jiang, W. Qi, H. Zhang, Y. Tang, Y. Hao, J. Yang, and M. Wang, “Loss crosstalk 1×2 thermooptic digital optical switch with integrated S-bend attenuator,” IEEE Photon. Technol. Lett. 18, 610-612 (2006).
[CrossRef]

Joyner, C. H.

M. Zirngibl, C. H. Joyner, L. W. Stulz, Th. Gaiffe, and C. Dragone, “Polarization independent 8×8 waveguide grating multiplexers on InP,” Electron. Lett. 29, 201-202 (1993).
[CrossRef]

Kabir, A. K. M. S.

B. M. A. Rahman, A. K. M. S. Kabir, M. Rajarajan, and K. T. V. Grattan, “Finite element modal solutions of planar photonic crystal fibers with rectangular air-holes,” Opt. Quantum Electron. 37, 171-183 (2005).
[CrossRef]

Kejalakshmy, N.

N. Kejalakshmy, B. M. A. Rahman, A. K. M. S. Kabir, M. Rajarajan, and K. T. V. Grattan, “Single mode operation of photonic crystal fiber using a full vectorial finite element method,” Proc. SPIE 6588, 65880T (2007).
[CrossRef]

Kim, S.

S. Kim and A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol. 14, 2085-2092 (1996).
[CrossRef]

Knight, J. C.

J. Arriaga, J. C. Knight, and P. St. J. Russell, “Modeling the propagation of light in photonic crystal fibers,” Physica D (Amsterdam) 189, 100-106 (2004).
[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]

Koshiba, M.

Lægsgaard, J.

J. Riishede, N. A. Mortensen, and J. Lægsgaard, “A “poor man's approach” to modelling micro-structured optical fibres,” J. Opt. A Pure Appl. Opt. 5, 534-538 (2003).
[CrossRef]

Lagasse, P. E.

Laine, J. P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

Lee, Y.

Leon-Saval, S.

Little, B. E.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

Lyytikainen, K.

Martynkien, T.

McPhedran, R. C.

Melchior, H.

J. Gu, P. Besse, and H. Melchior, “Novel method for analysis of curved optical rib-waveguides,” Electron. Lett. 25, 278-280(1989).
[CrossRef]

Monro, T. M.

Mortensen, N. A.

K. Saitoh, Y. Tsuchida, M. Koshiba, and N. A. Mortensen, “Endlessly single-mode holey fibers: the influence of core design,” Opt. Express 13, 10833-10839 (2005).
[CrossRef] [PubMed]

J. Riishede, N. A. Mortensen, and J. Lægsgaard, “A “poor man's approach” to modelling micro-structured optical fibres,” J. Opt. A Pure Appl. Opt. 5, 534-538 (2003).
[CrossRef]

Nasilowski, T.

Obayya, S. S. A.

M. Rajarajan, S. S. A. Obayya, B. M. A. Rahman, K. T. V. Grattan, and H. A. El-Mikathi, “Characterization of low-loss waveguide bends with offset optimisation for compact photonic integrated circuits,” IEE Proc. Optoelectron. 147, 382-388 (2000).
[CrossRef]

Olszewski, J.

Pierre, R.

Qi, W.

X. Jiang, W. Qi, H. Zhang, Y. Tang, Y. Hao, J. Yang, and M. Wang, “Loss crosstalk 1×2 thermooptic digital optical switch with integrated S-bend attenuator,” IEEE Photon. Technol. Lett. 18, 610-612 (2006).
[CrossRef]

Rahman, B. M. A.

N. Kejalakshmy, B. M. A. Rahman, A. K. M. S. Kabir, M. Rajarajan, and K. T. V. Grattan, “Single mode operation of photonic crystal fiber using a full vectorial finite element method,” Proc. SPIE 6588, 65880T (2007).
[CrossRef]

B. M. A. Rahman, A. K. M. S. Kabir, M. Rajarajan, and K. T. V. Grattan, “Finite element modal solutions of planar photonic crystal fibers with rectangular air-holes,” Opt. Quantum Electron. 37, 171-183 (2005).
[CrossRef]

M. Rajarajan, S. S. A. Obayya, B. M. A. Rahman, K. T. V. Grattan, and H. A. El-Mikathi, “Characterization of low-loss waveguide bends with offset optimisation for compact photonic integrated circuits,” IEE Proc. Optoelectron. 147, 382-388 (2000).
[CrossRef]

B. M. A. Rahman and J. B. Davies, “Finite-element solution of integrated optical waveguides,” J. Lightwave Technol. 2, 682-688 (1984).
[CrossRef]

Rajarajan, M.

N. Kejalakshmy, B. M. A. Rahman, A. K. M. S. Kabir, M. Rajarajan, and K. T. V. Grattan, “Single mode operation of photonic crystal fiber using a full vectorial finite element method,” Proc. SPIE 6588, 65880T (2007).
[CrossRef]

B. M. A. Rahman, A. K. M. S. Kabir, M. Rajarajan, and K. T. V. Grattan, “Finite element modal solutions of planar photonic crystal fibers with rectangular air-holes,” Opt. Quantum Electron. 37, 171-183 (2005).
[CrossRef]

M. Rajarajan, S. S. A. Obayya, B. M. A. Rahman, K. T. V. Grattan, and H. A. El-Mikathi, “Characterization of low-loss waveguide bends with offset optimisation for compact photonic integrated circuits,” IEE Proc. Optoelectron. 147, 382-388 (2000).
[CrossRef]

Richardson, D. J.

Riishede, J.

J. Riishede, N. A. Mortensen, and J. Lægsgaard, “A “poor man's approach” to modelling micro-structured optical fibres,” J. Opt. A Pure Appl. Opt. 5, 534-538 (2003).
[CrossRef]

Roelens, M.

P. Bienstman, E. Six, M. Roelens, M. Vanwolleghem, and R. Baets, “Calculation of bending losses in dielectric waveguides using eigenmode expansion and perfectly matched layers,” IEEE Photon. Technol. Lett. 14, 164-166(2002).
[CrossRef]

Russell, P. St. J.

Ryan, T.

S. Kabir, A. K. M.

N. Kejalakshmy, B. M. A. Rahman, A. K. M. S. Kabir, M. Rajarajan, and K. T. V. Grattan, “Single mode operation of photonic crystal fiber using a full vectorial finite element method,” Proc. SPIE 6588, 65880T (2007).
[CrossRef]

Saccomandi, L.

Saitoh, K.

Shenoy, M. R.

Shuisheng, J.

Shuqin, L.

Six, E.

P. Bienstman, E. Six, M. Roelens, M. Vanwolleghem, and R. Baets, “Calculation of bending losses in dielectric waveguides using eigenmode expansion and perfectly matched layers,” IEEE Photon. Technol. Lett. 14, 164-166(2002).
[CrossRef]

Skorobogatiy, M.

Steel, M. J.

Stulz, L. W.

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Figures (12)

Fig. 1
Fig. 1

Variation of the total losses with the bending radius R for the quasi-TM modes.

Fig. 2
Fig. 2

Variation of the effective indices with the bending radius R.

Fig. 3
Fig. 3

Variation of the total losses with the bending radius R.

Fig. 4
Fig. 4

H x field profile of the H x 11 core mode along the X axis through the center of the core when R = 1460 μm .

Fig. 5
Fig. 5

H x field contour for H x c 1 cladding mode when R = 1460 μm .

Fig. 6
Fig. 6

H x field profile for the H x 11 core mode when R = 1430 μm .

Fig. 7
Fig. 7

H x field profile for the H x c 1 cladding mode when R = 1430 μm .

Fig. 8
Fig. 8

Variation of the effective indices with the bending radius for the quasi-TE modes.

Fig. 9
Fig. 9

Variation of the total losses with the bending radius R.

Fig. 10
Fig. 10

H y field profile along the X axis for the H y 11 core mode when R = 840 μm above the degeneration point.

Fig. 11
Fig. 11

H y field profile along the X axis for the H y 11 core when R = 830 μm below the degeneration point.

Fig. 12
Fig. 12

H y contour of the higher order cladding mode when R = 833 μm .

Equations (4)

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ω 2 = ( ( × H ) * · ε ^ 1 ( × H ) d Ω ) + ( ( η / ε o ) ( · H ) * ( · H ) d Ω ) H * · μ ^ H d Ω ,
n eq ( x , y ) = n ( x , y ) ( 1 + x R ) ,
n eq ( x , y ) = n ( x , y ) [ 1 + ( 1 χ ) ( x R ) ] = n ( x , y ) ( 1 + x R ' ) .
A eff = ( s | E t | 2 d x d y ) 2 ( s | E t | 4 d x d y ) ,

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