Abstract

We demonstrate that a combination of multipole and Bloch methods is well suited for calculating the modes of air core photonic crystal fibers. This includes determining the reflective properties of the cladding, which is a prerequisite for the modal calculations. We demonstrate that in the presence of absorption, the modal losses can be substantially smaller than in the corresponding bulk medium.

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References

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  1. J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, "Photonic bandgap guidance in optical fibres," Science 282, 1476-1478 (1998).
    [CrossRef] [PubMed]
  2. J. Broeng, S.E. Barkou, T. Sondergaard, and A. Bjarklev, "Analysis of air-guiding photonic band gap fibres," Opt. Lett. 21, 1547-1549 (2000).
  3. F. Brechet, J. Marcou, D. Pagnoux, P.Roy, "Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method," Opt. Fibre Technol. 6, 181-191 (2000).
    [CrossRef]
  4. A. Ferrando, E. Silvestre, J.J. Miret, P. Andres, M.V. Andres, "Vector description of higher order modes in photonic crystal fibres," J. Opt. Soc. Am. B 17, 1333-1340 (2000).
    [CrossRef]
  5. Tanya M. Monro, D.J. Richardson, N.G.R. Broderick and P.J. Bennet, "Modeling large air fraction holey optical fibers," J. Lightwave Technol. 18, 50-56 (2000).
    [CrossRef]
  6. B.J. Eggleton, P.S. Westbrook, C.A. White, C. Kerbage, R.S. Windeler and G.L. Burdge, "Claading-modd-resonances in air-silica microstructure optical fibers," J. Lightwave Technol. 18, 1084-1100 (2000).
    [CrossRef]
  7. T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, G. Renversez, C.M. de Sterke, L.C. Botten, "Multipole method for microstructured optical fibers," submitted J. Opt. Soc Am B.
  8. L.C. Botten, N.A. Nicorovici, R.C. McPhedran, A.A. Asatryan, C. Martijn de Sterke, and P.A. Robinson, " Formulation for electromagnetic propagation and scattering by stacked gratings of metallic and dielectric cylinders, part 1: formulation," J. Opt. Soc. Am. A 17, 2165-2176 (2000).
    [CrossRef]
  9. L.C. Botten, N.A. Nicorovici, R.C. McPhedran, A.A. Asatryan, C. Martijn de Sterke, and P.A. Robinson, "Formulation for electromagnetic propagation and scattering by stacked gratings of metallic and dielectric cylinders, part 2: properties and implementation," J. Opt. Soc. Am. A 17, 2177-2190 (2000).
    [CrossRef]
  10. B. Gralak, S.Enoch, G. Tayeb, "Anomalous refractive properties of photonic crystals," J. Opt. Soc. Am A 17, 1012 (2000).
    [CrossRef]
  11. L. C. Botten, N. A. Nicorovici, R. C. McPhedran, A. A. Asatryan, C. M. de Sterke, "Photonic band calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
    [CrossRef]
  12. T.P White, R.C. McPhedran, C.M. de Sterke, L.C. Botten, and M.J. Steel, "Confinement losses in microstructured optical fibres," Opt. Lett. 26, 488-490 (2001).
    [CrossRef]
  13. R. C. McPhedran, N. A. Nicorovici, L. C. Botten and Bao Ke-Da, "Green's functions, lattice sums, and Rayleigh's Identity for a dynamic scattering problem", ed. G. Papanicolaou, IMA Volumes in Mathematics and its Applications, Vol. 96 (Springer, New York, 1997)
  14. N. A. Nicorovici, R. C. McPhedran and L. C. Botten, "Photonic band gaps for arrays of perfectly conducting cylinders," Phys. Rev. E 52, 1135 (1995).
    [CrossRef]
  15. R.C. McPhedran, N.A. Nicorovici, L.C. Botten, and K.A. Grubits, "Lattice sums for gratings and arrays," J. Math. Phys. 41, 7808 (2000).
    [CrossRef]
  16. G. H. Smith, L. C. Botten, R. C. McPhedran and N. A. Nicorovici, "Cylinder gratings in conical diffraction," in preparation for Phys. Rev. E.
  17. F. W. J. Olver, "Bessel Functions of Integer Order," in Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (Dover, New York, 1972), pp. 355-433.

Other

J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, "Photonic bandgap guidance in optical fibres," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

J. Broeng, S.E. Barkou, T. Sondergaard, and A. Bjarklev, "Analysis of air-guiding photonic band gap fibres," Opt. Lett. 21, 1547-1549 (2000).

F. Brechet, J. Marcou, D. Pagnoux, P.Roy, "Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method," Opt. Fibre Technol. 6, 181-191 (2000).
[CrossRef]

A. Ferrando, E. Silvestre, J.J. Miret, P. Andres, M.V. Andres, "Vector description of higher order modes in photonic crystal fibres," J. Opt. Soc. Am. B 17, 1333-1340 (2000).
[CrossRef]

Tanya M. Monro, D.J. Richardson, N.G.R. Broderick and P.J. Bennet, "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, "Claading-modd-resonances in air-silica microstructure optical fibers," J. Lightwave Technol. 18, 1084-1100 (2000).
[CrossRef]

T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, G. Renversez, C.M. de Sterke, L.C. Botten, "Multipole method for microstructured optical fibers," submitted J. Opt. Soc Am B.

L.C. Botten, N.A. Nicorovici, R.C. McPhedran, A.A. Asatryan, C. Martijn de Sterke, and P.A. Robinson, " Formulation for electromagnetic propagation and scattering by stacked gratings of metallic and dielectric cylinders, part 1: formulation," J. Opt. Soc. Am. A 17, 2165-2176 (2000).
[CrossRef]

L.C. Botten, N.A. Nicorovici, R.C. McPhedran, A.A. Asatryan, C. Martijn de Sterke, and P.A. Robinson, "Formulation for electromagnetic propagation and scattering by stacked gratings of metallic and dielectric cylinders, part 2: properties and implementation," J. Opt. Soc. Am. A 17, 2177-2190 (2000).
[CrossRef]

B. Gralak, S.Enoch, G. Tayeb, "Anomalous refractive properties of photonic crystals," J. Opt. Soc. Am A 17, 1012 (2000).
[CrossRef]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, A. A. Asatryan, C. M. de Sterke, "Photonic band calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

T.P White, R.C. McPhedran, C.M. de Sterke, L.C. Botten, and M.J. Steel, "Confinement losses in microstructured optical fibres," Opt. Lett. 26, 488-490 (2001).
[CrossRef]

R. C. McPhedran, N. A. Nicorovici, L. C. Botten and Bao Ke-Da, "Green's functions, lattice sums, and Rayleigh's Identity for a dynamic scattering problem", ed. G. Papanicolaou, IMA Volumes in Mathematics and its Applications, Vol. 96 (Springer, New York, 1997)

N. A. Nicorovici, R. C. McPhedran and L. C. Botten, "Photonic band gaps for arrays of perfectly conducting cylinders," Phys. Rev. E 52, 1135 (1995).
[CrossRef]

R.C. McPhedran, N.A. Nicorovici, L.C. Botten, and K.A. Grubits, "Lattice sums for gratings and arrays," J. Math. Phys. 41, 7808 (2000).
[CrossRef]

G. H. Smith, L. C. Botten, R. C. McPhedran and N. A. Nicorovici, "Cylinder gratings in conical diffraction," in preparation for Phys. Rev. E.

F. W. J. Olver, "Bessel Functions of Integer Order," in Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (Dover, New York, 1972), pp. 355-433.

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Figures (4)

Fig. 1.
Fig. 1.

Geometry of the unit cell (defined by the fundamental translation vectors e1 and e2. Phase origins P 1 and P 2 and representations of the incoming (δ ±) and outgoing (f ±) plane wave trains is depicted.

Fig. 2.
Fig. 2.

Dispersion diagram for a hexagonal microstructured optical fibre overlaid with the light line (magenta curve) and the modal dispersion curve (red curve). The fibre data are: hole diameter d=4.026 µm, hole spacing Λ=5.7816 µm, central hole diameter dc =13.1 µm.

Fig. 3.
Fig. 3.

Wavelength variation of the absorption reduction factor for the fibre of Fig. 2.

Fig. 4.
Fig. 4.

Longitudinal components of the electric field, the magnetic field and the Poynting vector at the wavelength, λ=3.428µm, where the MOF of Fig. 2 has maximum reduction in material absorption. At this wavelength, the matrix index is ne =1.39+i0.0003.

Equations (18)

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E z = m [ A m E l J m ( k e r c l ) + B m E l H m ( 1 ) ( k e r c l ) ] e im arg ( r c l ) ,
E z = l = 1 N c m B m E l H m ( 1 ) ( k e r c l ) e im arg ( r c l ) + m A m E 0 J m ( k e r ) e im θ ,
A E l = j l 𝓗 l j B E j + 𝓙 l 0 A E 0 , with 𝓗 lj = [ 𝓗 nm lj ] , 𝓙 l 0 = [ 𝓙 nm l 0 ] ,
𝓗 nm lj = H n m ( 1 ) ( k e c lj e i ( n m ) arg ( c lj ) , 𝓙 l 0 = J m n ( k e c l ) e i ( m n ) ) arg ( c l ) ,
B E 0 = l = 1 N c 𝓙 0 l B E l , where 𝓙 0 l = [ 𝓙 nm 0 l ] , 𝓙 nm 0 l = J n m ( k e c l ) e i ( n m ) arg ( c l ) ,
B El = R EE , l A E l + R EH , l A H l , B H l = R HE , l A E l + R HH , l A H l .
R ˜ l = ( R EE , l R EH , l R HE , l R HH , l ) , 𝓡 = diag ( R ˜ l )
𝓐 = 𝓗 𝓑 + 𝓙 B 0 A ˜ 0 , B ˜ 0 = 𝓙 0 B 𝓑 , with 𝓙 B 0 = [ 𝓙 l 0 ] and 𝓙 0 B = [ 𝓙 0 l ] .
𝓜 𝓑 ( I 𝓡 𝓢 ) 𝓑 = 0 , where 𝓢 = 𝓗 ˜ + 𝓙 ˜ B 0 R ˜ 0 𝓙 ˜ 0 B .
M ˜ B ˜ ( I R ˜ S ˜ ) B ˜ = 0 , with S ˜ A = ( S A 0 0 S A ) .
S n A j l H n 0 lj e i k 0 · ( c j c l ) = j 0 H n ( 1 ) ( k c j ) e in arg ( c j ) e i k 0 · c j ,
E t = s ξ s 1 2 ( δ s E e i χ s y + f s E + e i χ s y ) R s E + ξ s 1 2 ( δ s H e i χ s y + f s H + e i χ s y ) R s H
y ̂ × K t = s ξ s 1 2 ( δ s E e i χ s y f s E + e i χ s y ) R s E + ξ s 1 2 ( δ s H e i χ s y f s H + e i χ s y ) R s H
F W ( 0 ) Δ = [ I + 2 ω ξ ˜ T K ˜ ( I ˜ R ˜ S ˜ G ) 1 R ˜ J ˜ ξ ˜ ] Δ ,
W ( 0 ) = ( T 1 ( 0 ) R 1 ( 0 ) R 2 ( 0 ) T 2 ( 0 ) ) ,
W F = 0 where W = [ T μ I R R T μ 1 I ] ,
𝓘 = 1 2 ( I I I I ) , we form W = 𝓘 W 𝓘 T = ( T ' + R c I is I is I T R c I ) ,
𝓋 i 1 T g i = 1 2 c g i where 𝓋 i = I + ( T R ) ( T ± R ) , ( i = 1 , 2 ) .

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