Asymptotic solutions of eigenmodes in slab waveguides terminated by perfectly matched layers

Jianxin Zhu; Ya Yan Lu

doi:10.1364/JOSAA.30.002090

Journal of the Optical Society of America A
Vol. 30,
Issue 10,
pp. 2090-2095
(2013)
•https://doi.org/10.1364/JOSAA.30.002090

Asymptotic solutions of eigenmodes in slab waveguides terminated by perfectly matched layers

Jianxin Zhu and Ya Yan Lu

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Jianxin Zhu and Ya Yan Lu, "Asymptotic solutions of eigenmodes in slab waveguides terminated by perfectly matched layers," J. Opt. Soc. Am. A 30, 2090-2095 (2013)

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Abstract

For numerical modeling of optical wave-guiding structures, perfectly matched layers (PMLs) are widely used to terminate the transverse variables of the waveguide. The PML modes are the eigenmodes of a waveguide terminated by PMLs, and they have found important applications in the mode matching method, the coupled mode theory, and so on. In this paper, we consider PML modes for two-dimensional slab waveguides. It is shown that the PML modes consist of perturbed propagating modes, perturbed leaky modes, and two infinite sequences of Berenger modes. High-order asymptotic solutions for the Berenger modes are derived using a systematic approach.

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Equations (33)

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Sequence	Exact $β_{m} / k_{0}$	R.E. of (20)
1	$1.43929148 + 6.63289452 i$	$0.21 \times 10^{- 1}$
1	$1.61322522 + 7.59678078 i$	$0.89 \times 10^{- 2}$
1	$1.78855145 + 8.51026440 i$	$0.37 \times 10^{- 2}$
1	$1.95999515 + 9.38560677 i$	$0.16 \times 10^{- 2}$
1	$2.13878608 + 10.23243809 i$	$0.63 \times 10^{- 3}$
1	$2.33151054 + 11.06866741 i$	$0.21 \times 10^{- 3}$
1	$2.52955914 + 11.90377903 i$	$0.70 \times 10^{- 4}$
1	$2.72842216 + 12.73636055 i$	$0.22 \times 10^{- 4}$
1	$2.92805862 + 13.56549835 i$	$0.71 \times 10^{- 5}$
2	$1.12165964 + 2.50906348 i$	$0.47 \times 10^{- 1}$
2	$1.49026048 + 3.96589470 i$	$0.12 \times 10^{- 2}$
2	$2.00763631 + 5.30041599 i$	$0.16 \times 10^{- 3}$
2	$2.54001164 + 6.61296082 i$	$0.62 \times 10^{- 4}$
2	$3.08547425 + 7.91153908 i$	$0.27 \times 10^{- 4}$
2	$3.64048886 + 9.20100880 i$	$0.13 \times 10^{- 4}$
2	$4.20275879 + 10.48410130 i$	$0.69 \times 10^{- 5}$

Exact $β / k_{0}$	R.E. of (20)
$1.96391106 + 5.32063246 i$	$0.11 \times 10^{- 1}$
$2.05728030 + 5.28033113 i$	$0.71 \times 10^{- 2}$
$2.53330898 + 6.60819817 i$	$0.27 \times 10^{- 2}$
$2.56865978 + 6.60561514 i$	$0.23 \times 10^{- 2}$
$3.09196258 + 7.90452147 i$	$0.71 \times 10^{- 3}$
$3.10386252 + 7.90639448 i$	$0.71 \times 10^{- 3}$
$3.65136361 + 9.19430542 i$	$0.19 \times 10^{- 3}$
$3.65505052 + 9.19561386 i$	$0.21 \times 10^{- 3}$
$4.21504927 + 10.47774410 i$	$0.49 \times 10^{- 4}$
$4.21612167 + 10.47832680 i$	$0.60 \times 10^{- 4}$
$4.78345537 + 11.75629902 i$	$0.11 \times 10^{- 4}$
$4.78375196 + 11.75651836 i$	$0.18 \times 10^{- 4}$
$5.35612345 + 13.03115463 i$	$0.20 \times 10^{- 5}$
$5.35620165 + 13.03122997 i$	$0.63 \times 10^{- 5}$

Sequence	Exact $β / k_{0}$	R.E. of (31)
1	$1.22010469 + 4.32303238 i$	$0.23 \times 10^{- 2}$
1	$1.40473378 + 5.28413486 i$	$0.46 \times 10^{- 3}$
1	$1.60215116 + 6.19549084 i$	$0.95 \times 10^{- 4}$
1	$1.80584180 + 7.08042812 i$	$0.15 \times 10^{- 4}$
1	$2.01256752 + 7.94794035 i$	$0.86 \times 10^{- 5}$
2	$1.54685795 + 3.00455475 i$	$0.39 \times 10^{- 2}$
2	$2.14756656 + 4.29343843 i$	$0.73 \times 10^{- 3}$
2	$2.75336146 + 5.56421614 i$	$0.21 \times 10^{- 3}$
2	$3.36158633 + 6.82692719 i$	$0.76 \times 10^{- 4}$
2	$3.97119999 + 8.08533478 i$	$0.33 \times 10^{- 4}$
2	$4.58168840 + 9.34117889 i$	$0.16 \times 10^{- 4}$
2	$5.19276524 + 10.59537601 i$	$0.87 \times 10^{- 5}$

Exact $β / k_{0}$	R.E. of (31)
$0.95506150 + 1.64228368 i$	$0.50 \times 10^{- 1}$
$0.97763562 + 1.67133546 i$	$0.67 \times 10^{- 1}$
$1.54965425 + 3.00401425 i$	$0.44 \times 10^{- 2}$
$1.54854381 + 3.00354362 i$	$0.41 \times 10^{- 2}$
$2.14972377 + 4.29245186 i$	$0.79 \times 10^{- 3}$
$2.14974236 + 4.29241705 i$	$0.78 \times 10^{- 3}$
$2.75550723 + 5.56318760 i$	$0.22 \times 10^{- 3}$
$2.75550866 + 5.56318760 i$	$0.22 \times 10^{- 3}$
$3.36372614 + 6.82588911 i$	$0.82 \times 10^{- 4}$
$3.97333694 + 8.08429241 i$	$0.36 \times 10^{- 4}$
$4.58382416 + 9.34013462 i$	$0.17 \times 10^{- 4}$
$5.19490004 + 10.59433079 i$	$0.93 \times 10^{- 5}$

Sequence	Exact $β_{m} / k_{0}$	R.E. of (20)
1	$1.43929148 + 6.63289452 i$	$0.21 \times 10^{- 1}$
1	$1.61322522 + 7.59678078 i$	$0.89 \times 10^{- 2}$
1	$1.78855145 + 8.51026440 i$	$0.37 \times 10^{- 2}$
1	$1.95999515 + 9.38560677 i$	$0.16 \times 10^{- 2}$
1	$2.13878608 + 10.23243809 i$	$0.63 \times 10^{- 3}$
1	$2.33151054 + 11.06866741 i$	$0.21 \times 10^{- 3}$
1	$2.52955914 + 11.90377903 i$	$0.70 \times 10^{- 4}$
1	$2.72842216 + 12.73636055 i$	$0.22 \times 10^{- 4}$
1	$2.92805862 + 13.56549835 i$	$0.71 \times 10^{- 5}$
2	$1.12165964 + 2.50906348 i$	$0.47 \times 10^{- 1}$
2	$1.49026048 + 3.96589470 i$	$0.12 \times 10^{- 2}$
2	$2.00763631 + 5.30041599 i$	$0.16 \times 10^{- 3}$
2	$2.54001164 + 6.61296082 i$	$0.62 \times 10^{- 4}$
2	$3.08547425 + 7.91153908 i$	$0.27 \times 10^{- 4}$
2	$3.64048886 + 9.20100880 i$	$0.13 \times 10^{- 4}$
2	$4.20275879 + 10.48410130 i$	$0.69 \times 10^{- 5}$

Abstract

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Tables (4)

Equations (33)

Journal of the Optical Society of America A