Adaptive mode solver for varying refractive index profile&#x2019;s waveguides with modified spectral element method

Jianxin Zhu; Hao Yang

doi:10.1364/JOSAB.34.000538

Journal of the Optical Society of America B
Vol. 34,
Issue 3,
pp. 538-545
(2017)
•https://doi.org/10.1364/JOSAB.34.000538

Adaptive mode solver for varying refractive index profile’s waveguides with modified spectral element method

Jianxin Zhu and Hao Yang

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Jianxin Zhu and Hao Yang, "Adaptive mode solver for varying refractive index profile’s waveguides with modified spectral element method," J. Opt. Soc. Am. B 34, 538-545 (2017)

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Table of Contents Category
- Fiber Optics and Optical Communications
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The topics in this list come from the Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: October 14, 2016
- Revised Manuscript: December 31, 2016
- Manuscript Accepted: January 7, 2017
- Published: February 6, 2017

Abstract

A mode solver based on the spectral element method with mesh adaptation is proposed to calculate the modal characteristics of a semivectorial field in open varying-index optical waveguides. General optical waveguides with varying refractive indices are studied for the transverse electric and transverse magnetic cases, where perfectly matched layers (PMLs) are used to truncate the unbounded waveguides. The optical field expanded by a suitable set of orthogonal basis points (Gauss–Lobatto–Legendre or Gauss–Chebyshev collocation points) through the spectral element method are all represented by processing of characterization of composite material through mesh adaptation. By this combined solver, a large number of accurate PML modes can be easily calculated. Our results are still found to be in good agreement with those produced by other numerical methods, but with more efficiency. Additionally, the relations of the PML modes’ distribution to the parameters of PMLs are analyzed through the pitchfork phenomenon in the spectrum.

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Equations (34)

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Exact $β / k_{0}$	MSEM	SEM	CSCM	QFEM
$0.82356117 + 2.01455627 i$	$2.16 e - 4$	$4.31 e - 4$	$2.27 e - 4$	$6.68 e - 3$
$0.91276883 + 1.06278116 i$	$2.23 e - 4$	$4.12 e - 4$	$2.33 e - 4$	$6.75 e - 3$
$1.12759962 + 0.03427689 i$	$2.20 e - 4$	$4.14 e - 4$	$2.38 e - 4$	$6.77 e - 3$
$2.03893891 + 1.22261834 i$	$3.56 e - 4$	$4.77 e - 4$	$3.88 e - 4$	$8.33 e - 3$
$1.02431753 + 4.02522338 i$	$3.66 e - 4$	$6.83 e - 4$	$4.12 e - 4$	$7.33 e - 3$
$2.15605836 + 0.14590736 i$	$4.58 e - 4$	$8.22 e - 4$	$5.02 e - 4$	$7.29 e - 3$
$2.42386654 + 0.04350345 i$	$4.33 e - 4$	$7.48 e - 4$	$4.86 e - 4$	$6.55 e - 3$
$2.68543252 + 1.85673502 i$	$4.77 e - 4$	$7.51 e - 4$	$4.69 e - 4$	$6.17 e - 3$
$1.23733285 + 5.32812008 i$	$4.41 e - 4$	$7.27 e - 4$	$5.74 e - 4$	$7.29 e - 3$
$1.31572931 + 7.27681534 i$	$4.95 e - 4$	$7.68 e - 4$	$6.05 e - 4$	$7.37 e - 3$
$3.18855145 + 1.52026442 i$	$5.48 e - 4$	$8.23 e - 4$	$6.12 e - 4$	$8.67 e - 3$
$3.27117126 + 0.52455999 i$	$5.23 e - 4$	$7.97 e - 4$	$6.37 e - 4$	$8.84 e - 3$
$3.68948291 + 2.19100804 i$	$5.76 e - 4$	$8.64 e - 4$	$6.33 e - 4$	$9.02 e - 3$
$3.73261744 + 0.91377093 i$	$6.12 e - 4$	$8.85 e - 4$	$6.43 e - 4$	$9.27 e - 3$

Exact $β / k_{0}$	MSEM	SEM	CSCM	QFEM
$0.78429557 + 2.46332873 i$	$2.74 e - 4$	$4.63 e - 4$	$3.28 e - 4$	$7.24 e - 3$
$0.82175824 + 1.12741158 i$	$2.21 e - 4$	$4.57 e - 4$	$3.01 e - 4$	$6.96 e - 3$
$1.12161970 + 0.02300095 i$	$3.18 e - 4$	$6.17 e - 4$	$4.18 e - 4$	$7.62 e - 3$
$1.02431795 + 4.33765226 i$	$3.02 e - 4$	$5.13 e - 4$	$3.57 e - 4$	$7.35 e - 3$
$1.12559962 + 0.05427689 i$	$2.89 e - 4$	$4.33 e - 4$	$2.01 e - 4$	$6.63 e - 3$
$1.73564379 + 0.21590910 i$	$2.35 e - 4$	$5.02 e - 4$	$2.67 e - 4$	$8.36 e - 3$
$2.07035833 + 0.18798357 i$	$5.03 e - 4$	$8.66 e - 4$	$5.12 e - 4$	$8.16 e - 3$
$1.35516892 + 8.78243193 i$	$4.28 e - 4$	$7.15 e - 4$	$4.72 e - 4$	$7.94 e - 3$
$2.17342166 + 0.03252129 i$	$4.35 e - 4$	$7.23 e - 4$	$3.46 e - 4$	$6.88 e - 3$
$2.08334271 + 0.02875331 i$	$4.17 e - 4$	$6.96 e - 4$	$4.26 e - 4$	$7.31 e - 3$
$2.35631809 + 0.42386437 i$	$4.17 e - 4$	$8.35 e - 4$	$4.54 e - 4$	$7.28 e - 3$
$3.28502641 + 0.16754023 i$	$5.45 e - 4$	$8.86 e - 4$	$5.13 e - 4$	$9.12 e - 3$
$3.54245869 + 0.17051588 i$	$5.95 e - 4$	$8.78 e - 4$	$5.84 e - 4$	$1.03 e - 2$
$3.32163312 + 1.11460249 i$	$6.35 e - 4$	$7.23 e - 3$	$6.12 e - 4$	$1.69 e - 2$

Exact $β / k_{0}$	MSEM	SEM	CSCM	QFEM
$0.82356117 + 2.01455627 i$	$2.16 e - 4$	$4.31 e - 4$	$2.27 e - 4$	$6.68 e - 3$
$0.91276883 + 1.06278116 i$	$2.23 e - 4$	$4.12 e - 4$	$2.33 e - 4$	$6.75 e - 3$
$1.12759962 + 0.03427689 i$	$2.20 e - 4$	$4.14 e - 4$	$2.38 e - 4$	$6.77 e - 3$
$2.03893891 + 1.22261834 i$	$3.56 e - 4$	$4.77 e - 4$	$3.88 e - 4$	$8.33 e - 3$
$1.02431753 + 4.02522338 i$	$3.66 e - 4$	$6.83 e - 4$	$4.12 e - 4$	$7.33 e - 3$
$2.15605836 + 0.14590736 i$	$4.58 e - 4$	$8.22 e - 4$	$5.02 e - 4$	$7.29 e - 3$
$2.42386654 + 0.04350345 i$	$4.33 e - 4$	$7.48 e - 4$	$4.86 e - 4$	$6.55 e - 3$
$2.68543252 + 1.85673502 i$	$4.77 e - 4$	$7.51 e - 4$	$4.69 e - 4$	$6.17 e - 3$
$1.23733285 + 5.32812008 i$	$4.41 e - 4$	$7.27 e - 4$	$5.74 e - 4$	$7.29 e - 3$
$1.31572931 + 7.27681534 i$	$4.95 e - 4$	$7.68 e - 4$	$6.05 e - 4$	$7.37 e - 3$
$3.18855145 + 1.52026442 i$	$5.48 e - 4$	$8.23 e - 4$	$6.12 e - 4$	$8.67 e - 3$
$3.27117126 + 0.52455999 i$	$5.23 e - 4$	$7.97 e - 4$	$6.37 e - 4$	$8.84 e - 3$
$3.68948291 + 2.19100804 i$	$5.76 e - 4$	$8.64 e - 4$	$6.33 e - 4$	$9.02 e - 3$
$3.73261744 + 0.91377093 i$	$6.12 e - 4$	$8.85 e - 4$	$6.43 e - 4$	$9.27 e - 3$

Exact $β / k_{0}$	MSEM	SEM	CSCM	QFEM
$0.78429557 + 2.46332873 i$	$2.74 e - 4$	$4.63 e - 4$	$3.28 e - 4$	$7.24 e - 3$
$0.82175824 + 1.12741158 i$	$2.21 e - 4$	$4.57 e - 4$	$3.01 e - 4$	$6.96 e - 3$
$1.12161970 + 0.02300095 i$	$3.18 e - 4$	$6.17 e - 4$	$4.18 e - 4$	$7.62 e - 3$
$1.02431795 + 4.33765226 i$	$3.02 e - 4$	$5.13 e - 4$	$3.57 e - 4$	$7.35 e - 3$
$1.12559962 + 0.05427689 i$	$2.89 e - 4$	$4.33 e - 4$	$2.01 e - 4$	$6.63 e - 3$
$1.73564379 + 0.21590910 i$	$2.35 e - 4$	$5.02 e - 4$	$2.67 e - 4$	$8.36 e - 3$
$2.07035833 + 0.18798357 i$	$5.03 e - 4$	$8.66 e - 4$	$5.12 e - 4$	$8.16 e - 3$
$1.35516892 + 8.78243193 i$	$4.28 e - 4$	$7.15 e - 4$	$4.72 e - 4$	$7.94 e - 3$
$2.17342166 + 0.03252129 i$	$4.35 e - 4$	$7.23 e - 4$	$3.46 e - 4$	$6.88 e - 3$
$2.08334271 + 0.02875331 i$	$4.17 e - 4$	$6.96 e - 4$	$4.26 e - 4$	$7.31 e - 3$
$2.35631809 + 0.42386437 i$	$4.17 e - 4$	$8.35 e - 4$	$4.54 e - 4$	$7.28 e - 3$
$3.28502641 + 0.16754023 i$	$5.45 e - 4$	$8.86 e - 4$	$5.13 e - 4$	$9.12 e - 3$
$3.54245869 + 0.17051588 i$	$5.95 e - 4$	$8.78 e - 4$	$5.84 e - 4$	$1.03 e - 2$
$3.32163312 + 1.11460249 i$	$6.35 e - 4$	$7.23 e - 3$	$6.12 e - 4$	$1.69 e - 2$

Abstract

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Tables (2)

Equations (34)

Journal of the Optical Society of America B