Abstract

The paper presents a fully vectorial analysis of bending losses in photonic crystal fibers employing edge/nodal hybrid elements and perfectly matched layers boundary conditions. The oscillatory character of losses vs. both the wavelength and the bending radius has been demonstrated. The shown oscillations originate from the coupling between the fundamental mode guided in the core and the gallery of cladding modes arising due to light reflection from the boundary between solid and holey part of the cladding.

© 2005 Optical Society of America

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References

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Appl. Opt.

Electromagnetics

Y. Tsuji, M. Koshiba, �??Complex modal analysis of curved optical waveguides using a full-vectorial finite element method with perfectly matched layer boundary conditions,�?? Electromagnetics 24, 39-48 (2004).
[CrossRef]

Electron. Lett.

T. Sørensen, J. Broeng, A. Bjarklev, E. Knudsen, and S. E. B. Libori, �??Macro-bending loss properties of photonic crystal fibre,�?? Electron. Lett. 37, 287�??289 (2001).
[CrossRef]

IEE J. Quant. Electron.

M. Heiblum, J. H. Harris �??Analysis of curved optical waveguides by conformal transformation,�?? IEE J. Quant. Electron. 11, 75-83 (1975).
[CrossRef]

IEEE Microwave Guided Wave Lett.

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

J. Lightwave Technol.

J. H. Harris and P. F. Castle �??Bend loss measurements on high numerical aperture single-mode fibers as a function of wavelength and bend radius,�?? J. Lightwave Technol. 4, 34-40 (1986).
[CrossRef]

H. Renner, �??Bending losses of coated single-mode fibers: a simple approach,�?? J. Lightwave Technol. 10, 544-551 (1992).
[CrossRef]

L. Faustini, and G. Martini, �??Bend loss in single-mode fibers ,�?? J. Lightwave Technol. 15, 671-679 (1997).
[CrossRef]

T. M. Monro, D. J. Richardson, N. G. R. Broderick, P. J. Bennet, �??Holey optical fibers an efficient modal model,�?? J. Lightwave Technol. 17, 1093-1102 (1999).
[CrossRef]

Opt. Commun

M. Koshiba, K. Saitoh, �??Simple evaluation of confinement losses in holey fibers�?? Opt. Commun., article in press, (2005).
[CrossRef]

Opt. Commun.

J. C. Baggett , T. M. Monro, K. Furusawa,V. Finazzi, D.J. Richardson. �??Understanding bending losses in holey optical fibers,�?? Opt. Commun. 227, 317�??335 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

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Figures (6)

Fig. 1.
Fig. 1.

Computational domain limited with rectangular PML for the fiber bent in the xz-(a) and yz-plane (b), z-axis overlaps with the fiber symmetry axis. The geometrical parameters of the analyzed fiber are: pitch distance Λ = 13.2 μm and fill factor d/Λ= 0.485.

Fig. 2.
Fig. 2.

Bending losses for the y-polarized fundamental mode calculated versus bending radius using the fully vectorial method. The calculations were performed for the xz (a) and yz bending plane (b) at λ = 0.83 μm. Green line indicates losses determined using simplified analytical formula from ref. [5]. Solid lines 1, 4 and 9 correspond to the cladding modes with m = 1 symmetry arising for xz-bent.

Fig. 3.
Fig. 3.

Bending losses for the y-polarized fundamental mode calculated versus wavelength using the fully vectorial method. The calculations were performed for the xz (a) and yz bending plane (b) at R = 80 mm. Green line indicates losses determined using simplified analytical formula from ref. [5].

Fig. 4.
Fig. 4.

Distribution of the dominant component of the electric vector in selected y-polarized cladding modes. The calculations were performed for the xz (a) and yz-bending plane (b) for radii assuring peak losses, λ = 0.83 μm. Modes’ numeration is the same as in Fig. 2.

Fig. 5.
Fig. 5.

Effective indices of the fundamental mode (nf ) and the cladding modes calculated vs. bending radius for bent in the xz-plane (a) and yz-plane (b), λ = 0.83 μm. Modes’ numeration is the same as in Fig. 2.

Fig. 6.
Fig. 6.

Intensity distribution in the fundamental mode (lower half of the fiber cross-section) and in the cladding modes number 9 and 6 (upper half) calculated respectively for bent in the xz-plane (a) and yz-plane (b). The calculations were carried out for λ = 0.83 μm and radii assuring peak losses. A logarithmic scale with a step of 1.5 dB between successive contour lines and independent normalization of both intensity distributions to 0 dB were used to better show the overlap regions.

Equations (12)

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n eq x y = n x y exp ( p R ) ,
× ( [ μ ] PML 1 × E ̿ ) k 0 2 [ ε ] PML E ̿ = 0 ,
E ( r ) ̿ = [ S ] 1 E ( r ̄ ) ,
r ̄ = [ 0 x s x ( x ) dx , 0 y s y ( y ) dy , 0 z s z ( z ) dz ] T ,
[ S ] = [ S x 1 0 0 0 S y 1 0 0 0 S z 1 ] ,
[ μ ] PML = det 1 ( [ S ] ) [ S ] [ μ ( r ̄ ) ] [ S ] ,
[ ε ] PML = det 1 ( [ S ] ) [ S ] n 2 ( r ̄ ) [ S ] ,
s z = 1 , s y = 1 , s x ( x ) = { 1 for x x PML 1 ( x x PML d PML ) 2 for x > x PML ,
s z = 1 , s x = 1 , s y ( y ) = { 1 for y y PML 1 ( y y PML d PML ) 2 for y > y PML ,
n ( r ̄ ) = n eq ( r ) ,
n ( r ̄ ) = n eq ( r ̄ ) .
L B = 20 ln ( 10 ) Im ( β )

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