Calculations of air-guided modes in photonic crystal fibers using the multipole method

T.P. White; R.C. McPhedran; L.C. Botten; G.H. Smith; C. Martijn de Sterke

doi:10.1364/OE.9.000721

Optics Express
Vol. 9,
Issue 13,
pp. 721-732
(2001)
•https://doi.org/10.1364/OE.9.000721

Calculations of air-guided modes in photonic crystal fibers using the multipole method

T.P. White, R.C. McPhedran, L.C. Botten, G.H. Smith, and C. Martijn de Sterke

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T.P. White, R.C. McPhedran, L.C. Botten, G.H. Smith, and C. Martijn de Sterke, "Calculations of air-guided modes in photonic crystal fibers using the multipole method," Opt. Express 9, 721-732 (2001)

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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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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.

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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 d_c =13.1 µm.

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Fig. 3. Wavelength variation of the absorption reduction factor for the fibre of Fig. 2.

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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 n_e =1.39+i0.0003.

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Equations (18)

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E_{z} = \sum_{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} = \sum_{l = 1}^{N_{c}} \sum_{m} B_{m}^{E l} H_{m}^{(1)} (k_{⊥}^{e} ∣ r - c_{l} ∣) e^{im \arg (r - c_{l})} + \sum_{m} A_{m}^{E 0} J_{m} (k_{⊥}^{e} r) e^{im θ},

A^{E l} = \sum_{j \neq 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} = \sum_{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} .

{\tilde{R}}^{l} = (\begin{matrix} R^{EE, l} & R^{EH, l} \\ R^{HE, l} & R^{HH, l} \end{matrix}), 𝓡 = diag ({\tilde{R}}^{l})

𝓐 = 𝓗 𝓑 + 𝓙^{B 0} {\tilde{A}}^{0}, {\tilde{B}}^{0} = 𝓙^{0 B} 𝓑, with 𝓙^{B 0} = [𝓙^{l 0}] and 𝓙^{0 B} = [𝓙^{0 l}] .

𝓜 𝓑 \equiv (I - 𝓡 𝓢) 𝓑 = 0, where 𝓢 = \tilde{𝓗} + {\tilde{𝓙}}^{B 0} {\tilde{R}}^{0} {\tilde{𝓙}}^{0 B} .

\tilde{M} \tilde{B} \equiv (I - \tilde{R} \tilde{S}) \tilde{B} = 0, with {\tilde{S}}^{A} = (\begin{matrix} S^{A} & 0 \\ 0 & S^{A} \end{matrix}) .

S_{n}^{A} \equiv \sum_{j \neq l} H_{n 0}^{lj} e^{i k_{0} \cdot (c_{j} - c_{l})} = \sum_{j \neq 0} H_{n}^{(1)} (k ∣ c_{j} ∣) e^{- in \arg (c_{j})} e^{i k_{0} \cdot c_{j}},

E_{t} = \sum_{s} ξ_{s}^{- \frac{1}{2}} (δ_{s}^{E -} e^{- i χ_{s} y} + f_{s}^{E +} e^{i χ_{s} y}) R_{s}^{E} + ξ_{s}^{\frac{1}{2}} (δ_{s}^{H -} e^{- i χ_{s} y} + f_{s}^{H +} e^{i χ_{s} y}) R_{s}^{H}

\hat{y} \times K_{t} = \sum_{s} ξ_{s}^{\frac{1}{2}} (δ_{s}^{E -} e^{- i χ_{s} y} - f_{s}^{E +} e^{i χ_{s} y}) R_{s}^{E} + ξ_{s}^{- \frac{1}{2}} (δ_{s}^{H -} e^{- i χ_{s} y} - f_{s}^{H +} e^{i χ_{s} y}) R_{s}^{H}

F \equiv W^{(0)} Δ = [I + 2 ω {\tilde{ξ}}^{T} \tilde{K} {(\tilde{I} - \tilde{R} {\tilde{S}}^{G})}^{- 1} \tilde{R} \tilde{J} \tilde{ξ}] Δ,

W^{(0)} = (\begin{matrix} T_{1}^{(0)} & R_{1}^{(0)} \\ R_{2}^{(0)} & T_{2}^{(0)} \end{matrix}),

W' F' = 0 where W' = [\begin{matrix} T' - μ' I & R' \\ R' & T' - {μ'}^{- 1} I \end{matrix}],

𝓘 = \frac{1}{\sqrt{2}} (\begin{matrix} I & I \\ - I & I \end{matrix}), we form W^{†} = 𝓘 W' 𝓘^{T} = (\begin{matrix} T' + R' - c I & is I \\ is I & T' - R' - c I \end{matrix}),

𝓋_{i}^{- 1} T_{g i} = \frac{1}{2 c} g_{i} where 𝓋_{i} = I + (T \mp R) (T \pm R), (i = 1, 2) .

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Equations (18)

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