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A three-dimensional wide-angle BPM for optical waveguide structures

Changbao Ma and Edward Van Keuren

Open Access Open Access

Abstract

Algorithms for effective modeling of optical propagation in three- dimensional waveguide structures are critical for the design of photonic devices. We present a three-dimensional (3-D) wide-angle beam propagation method (WA-BPM) using Hoekstra’s scheme. A sparse matrix algebraic equation is formed and solved using iterative methods. The applicability, accuracy and effectiveness of our method are demonstrated by applying it to simulations of wide-angle beam propagation, along with a technique for shifting the simulation window to reduce the dimension of the numerical equation and a threshold technique to further ensure its convergence. These techniques can ensure the implementation of iterative methods for waveguide structures by relaxing the convergence problem, which will further enable us to develop higher-order 3-D WA-BPMs based on Padé approximant operators.

©2007 Optical Society of America

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Tables (1)

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Table 1. The relative L2 norm errors and relative position shift

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Equations (11)

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∂ 2 E ∂ x 2 + ∂ 2 E ∂ y 2 + ∂ 2 E ∂ z 2 + k 0 2 n 2 x y z E = 0
E x y z = ψ x y z exp ( − ik 0 n 0 z )
ia ∂ ψ ∂ − ∂ 2 ψ ∂ z 2 = ∂ 2 ψ ∂ x 2 + ∂ 2 ψ ∂ y 2 + b ψ
ia ∂ ψ ∂ z = ∂ 2 ψ ∂ x 2 + ∂ 2 ψ ∂ y 2 + b ψ
ia ψ l + 1 − ψ l Δz − 1 Δz [ ∂ ψ ∂ z l + 1 − ∂ ψ ∂ z l ]
= ∂ 2 ∂ x 2 ( ψ l + 1 + ψ l 2 ) + ∂ 2 ∂ y 2 ( ψ l + 1 + ψ l 2 ) + ( b ψ ) l + 1 + ( b ψ ) l 2
∂ ψ ∂ z l = 1 ia [ ∂ 2 ψ l ∂ x 2 + ∂ 2 ψ l ∂ y 2 + ( b ψ ) l ]
∂ ψ ∂ z l + 1 = 1 ia [ ∂ 2 ψ l + 1 ∂ x 2 + ∂ 2 ψ l + 1 ∂ y 2 + ( b ψ ) l + 1 ]
A ψ m − 1 , j l + 1 + B ψ m , j − 1 l + 1 + C ψ m , j l + 1 + D ψ m , j + 1 l + 1 + E ψ m + 1 , j l + 1
= R ψ m − 1 , j l + S ψ m , j − 1 l + T ψ m , j l + U ψ m , j + 1 l + V ψ m + 1 + j l
{ A = E = 1 Δ x 2 − 1 + 2 i a Δ z B = D = 1 Δ y 2 − 1 + 2 i a Δ z C = 1 Δ x 2 + 1 Δ y 2 2 − 4 i a Δ z + 2 ia Δ z − 1 − 2 i a Δ z b m , j l + 1 R = V = 1 Δ x 2 + 1 + 2 i a Δ z S = U = 1 Δ y 2 + 1 + 2 i a Δ z T = 1 Δ x 2 + 1 Δ y 2 − 2 − 4 i a Δ z + 2 ia Δ z + 1 + 2 i a Δ z b m , j i
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