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Bloch method for the analysis of modes in microstructured optical fibers

Boris T. Kuhlmey, Ross C. McPhedran, and C. Martijn de Sterke

Open Access Open Access

Abstract

We discuss a transform technique for analyzing the wave vector content of microstructured optical fiber (MOF) modes, which is computationally efficient and gives good physical insight into the nature of the mode. In particular, if the mode undergoes a transition from a bound state to an extended state, this is evident in the spreading-out of its transform. The method has been implemented in the multipole formulation for finding MOF modes, but are capable of adaptation to other formulations.

©2004 Optical Society of America

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Figures (2)
Fig. 1.
Fig. 1. Field maps and total Bloch transform of a mode consisting essentially of a superposition of 6 Bloch waves. Note that the fields are depicted in the direct space (r-space), whereas the Bloch transform is in the reciprocal space (k-space): the white hexagon on the Bloch transform map depicts the edges of the first Brillouin zone. Here Λ=2.3 µm, λ=1.55 µm, d/Λ=0.15, and n silica=1.44402036.

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Fig. 2.
Fig. 2. Fundamental mode of two MOFs with different pitch, but with same d/Λ=0.3 and N r=8. The field distribution changes considerably between the two values of the pitch, but the Bloch transform remains a single peak centered on the origin. For all figures λ=1.55 µm and n silica=1.44402036.

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Equations (14)

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𝓑 n ( k ) = ∑ l = 1 N i exp ( − i k · c l ) B n ( c l ) .
V ( r ) = ∑ m = 1 N B exp ( i k B m · r ) v k B m ( r ) ,
B n ( c l ) = ∑ m = 1 N B B ̂ n m exp ( i k B m · c l ) ,
∀ c ∈ 𝓛 , G · c ∈ 2 π ℤ ,
𝓑 T ( k ) = ∑ n 1 sup k ′ ∈ ℝ 2 ( ∣ 𝓑 n ( k ′ ) ∣ ) ∣ 𝓑 n ( k ) ∣ .
𝓑 n ( k B m ) = ∑ l = 1 N i ∑ j = 1 N B B ̂ n j exp ( i ( k B m − k B j ) · c l ) ,
= N i B ̂ n m + ∑ j = 1 , j ≠ m N B B ̂ n j ∑ l = 1 N i exp ( i ( k B m − k B j ) · c l ) .
∀ G ∈ 𝓛 * , ∀ j ≠ m , ∣ k B m − k B j + G ∣ N i 1 ⁄ 2 Λ ≳ 4 .
𝓑 n ( k B m ) ≃ N i B ̂ n m .
B ̂ n m ( Σ j = 1 N B ∣ B ̂ n j ∣ 2 ) 1 ⁄ 2 .
∑ l = 1 N i ∣ B n ( c l ) ∣ 2 = 1 𝓐 FBZ ∫ ∫ FBZ ∣ 𝓑 n ( k ) ∣ 2 d k ≃ N i ∑ i = 1 N B ∣ B ̂ n i ∣ 2 ,
∣ 𝓑 n ( k ) ∣ = ∣ B ̂ n ∣ ∣ sin ( N 1 ( k − k b ) · u 1 Λ ⁄ 2 ) sin ( ( k − k b ) · u 1 Λ ⁄ 2 ) ∣ ∣ sin ( N 2 ( k − k b ) · u 2 Λ ⁄ 2 ) sin ( ( k − k b ) · u 2 Λ ⁄ 2 ) ∣ ,
f ( x ) = { ∣ sin ( a x ) sin ( x ) ∣ if x ≠ m π , m ∈ ℤ ∣ a ∣ if x = m π , m ∈ ℤ .
δ k m ≃ 2 1.91 N m Λ .
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