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Analysis of the space filling modes of photonic crystal fibers

Zhaoming Zhu and Thomas G. Brown

Open Access Open Access

Abstract

We study the cladding modes of photonic crystal fibers (PCFs) using a fully vectorial method. This approach enables us to analyze the modes and incorporate material dispersion in a straightforward fashion. We find the field flow lines, intensity distribution and polarization properties of these modes. The effective cladding indices of different PCFs are investigated in detail.

©2001 Optical Society of America

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Figures (7)
Fig. 1.
Fig. 1. Schematic of the triangular lattice of air holes in the cladding. The lattice pitch is Λ, and R is the radius of the air holes. Dielectric constants of air and silica are εa =1.0 and εb (λ), respectively.

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Fig. 2.
Fig. 2. Dispersions of the some lower cladding modes. The fundamental mode (FCM) is a two-fold degenerate mode. The next lower mode (HCM) includes six nearly degenerate modes. The material dispersion of the silica core is also shown in the figure.

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Fig. 3.
Fig. 3. Transverse magnetic fields of the two degenerate modes of the fundamental cladding mode.

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Fig. 4.
Fig. 4. Transverse magnetic field intensity of the fundamental cladding mode (x-polarized).

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Fig. 5.
Fig. 5. Dispersion curves calculated using the present vectorial method (solid lines) and the scalar method (dashed lines) for claddings of different air-hole radius R.

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Fig. 6.
Fig. 6. Effective cladding index as a function of λ/Λ and R/Λ. Λ is chosen to be 2.3 µm and the material dispersion of silica is considered in the calculations.

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Fig. 7.
Fig. 7. Calculated parameter V vs. wavelength for claddings of different pitches and air-hole radii. The red dashed line indicates the single-mode cutoff value of V (=2.405). (a) Lattice pitch Λ=2.3 µm. Results from the vectorial (scalar) method are shown by blue (black) solid lines. (b) Lattice pitch Λ=5.0 µm. Only the vectorial results are shown.

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Equations (14)

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H ( x , y , z ; t ) = [ H t ( x , y ) + H z ( x , y ) z ̂ ] exp ( i β z − i ω t )
∇ 2 H + k 2 ε H = − ∇ ( ln ε ) × ( ∇ × H ) ,
LH t = ( ∇ t 2 + k 2 ε ) H t + ∇ t ( ln ε ) × ( ∇ t × H t ) = β 2 H t ,
ε ( x , y ) = ∑ G ε ̂ ( G ) exp ( i G · x t ) ,
ln ε ( x , y ) = ∑ G κ ̂ ( G ) exp ( i G · x t ) ,
H j ( x , y ) = ∑ G H ̂ j ( G ) exp ( i G · x t ) , j = x , y .
∑ G ′ L G ′ G H ̂ ( G ′ ) = β 2 H ̂ ( G ) ,
[ L G ′ G ] u , v = [ − ∣ G ∣ 2 δ G ′ , G + k 2 ∑ G ″ ε ̂ ( G ″ ) ] δ u , v + [ Q G ′ G ] u , v , ( u , v = x , y )
[ Q G ′ G ] x , x = κ ̂ ( G − G ′ ) ( G y − G y ′ ) G y ′ ,
[ Q G ′ G ] x , y = − κ ̂ ( G − G ′ ) ( G y − G y ′ ) G x ′ ,
[ Q G ′ G ] y , x = − κ ̂ ( G − G ′ ) ( G x − G x ′ ) G y ′ ,
[ Q G ′ G ] y , y = κ ̂ ( G − G ′ ) ( G x − G x ′ ) G x ′ .
ε ̂ ( G ) = { ε b + ( ε a − ε b ) f , G = 0 ( ε a − ε b ) f 2 J 1 ( GR ) GR , G ≠ 0
κ ̂ ( G ) = { ln ε b + ( ln ε a − ln ε b ) f , G = 0 ( ln ε a − ln ε b ) f 2 J 1 ( GR ) GR , G ≠ 0 ,
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