Variational approach for extraction of eigenmodes in layered waveguides

Khashayar Mehrany; Sina Khorasani; Bizhan Rashidian

doi:10.1364/JOSAB.19.001978

Journal of the Optical Society of America B
Vol. 19,
Issue 9,
pp. 1978-1981
(2002)
•https://doi.org/10.1364/JOSAB.19.001978

Variational approach for extraction of eigenmodes in layered waveguides

Khashayar Mehrany, Sina Khorasani, and Bizhan Rashidian

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Khashayar Mehrany, Sina Khorasani, and Bizhan Rashidian, "Variational approach for extraction of eigenmodes in layered waveguides," J. Opt. Soc. Am. B 19, 1978-1981 (2002)

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Abstract

A new variational approach to the extraction of guided and leaky modes in layered waveguides is proposed. To verify the method we compare the results of analysis of a typical test case and those from other references and find them in agreement. The efficiency of the proposed approach is compared with that of other reported methods.

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Mode	Method
	APMb	ADRc	ADRd	Variational
	APMb	ADRc	ADRd	With $m = 2$	With $m = 5$
${TE}_{0}$	1.6227286	1.6227287	1.622729	1.622728682	1.622728682
${TE}_{1}$	1.6052756	1.6052757	1.605276	1.605275698	1.605275698
${TE}_{2}$	1.5571361	1.5571361	1.557136	1.557136152	1.557136152
${TE}_{3}$	1.5035871	1.5035870	1.503587	1.503587112	1.503587112
${TM}_{0}$	1.6200313	e	1.620031	1.620031318	1.620031318
${TM}_{1}$	1.5947884	e	1.594788	1.594788478	1.594788478
${TM}_{2}$	1.5549806	e	1.554981	1.554980690	1.554980690
${TM}_{3}$	1.5018178	e	1.501818	1.501817805	1.501817805

Mode	Method
Mode	APMb	ADRc	ADRd	RPMe	Variational with Either $m = 2$ or $m = 5$
${TE}_{4}$	1.461857	1.4618566	1.46186	1.46185	1.461856641
${TE}_{4}$	$- j 7.1558 \times 10^{- 3}$	$- j 7.156 \times 10^{- 3}$	$- j 7.16 \times 10^{- 3}$	$- j 7 \times 10^{- 3}$	$- j 7.155870649 \times 10^{- 3}$
${TE}_{5}$	1.382489	1.3824892	1.38250	1.38239	1.382489223
${TE}_{5}$	$- j 1.8166 \times 10^{- 2}$	$- j 1.8166 \times 10^{- 2}$	$- j 1.817 \times 10^{- 2}$	$- j 1.72 \times 10^{- 2}$	$- j 1.816587736 \times 10^{- 2}$
${TE}_{6}$	1.281364	1.2813644	1.28136	1.28135	1.281364436
${TE}_{6}$	$- j 3.5877 \times 10^{- 2}$	$- j 3.5877 \times 10^{- 2}$	$- j 3.588 \times 10^{- 2}$	$- j 3.30 \times 10^{- 2}$	$- j 3.587739216 \times 10^{- 2}$
${TE}_{7}$	1.142314	1.1423144	1.14231	1.14155	1.142314462
${TE}_{7}$	$- j 5.2876 \times 10^{- 2}$	$- j 5.2876 \times 10^{- 2}$	$- j 5.288 \times 10^{- 2}$	$- j 4.80 \times 10^{- 2}$	$- j 5.287607512 \times 10^{- 2}$
${TE}_{8}$	1.003037	1.0030370	1.00304	1.00304	1.003037019
	$- j 7.0771 \times 10^{- 2}$	$- j 7.0771 \times 10^{- 2}$	$- j 7.077 \times 10^{- 2}$	$- j 6.56 \times 10^{- 1}$	$- j 7.077094109 \times 10^{- 2}$

Mode	Method
	APMb	ADRc	ADRd	Variational
	APMb	ADRc	ADRd	With $m = 2$	With $m = 5$
${TE}_{0}$	1.6227286	1.6227287	1.622729	1.622728682	1.622728682
${TE}_{1}$	1.6052756	1.6052757	1.605276	1.605275698	1.605275698
${TE}_{2}$	1.5571361	1.5571361	1.557136	1.557136152	1.557136152
${TE}_{3}$	1.5035871	1.5035870	1.503587	1.503587112	1.503587112
${TM}_{0}$	1.6200313	e	1.620031	1.620031318	1.620031318
${TM}_{1}$	1.5947884	e	1.594788	1.594788478	1.594788478
${TM}_{2}$	1.5549806	e	1.554981	1.554980690	1.554980690
${TM}_{3}$	1.5018178	e	1.501818	1.501817805	1.501817805

Mode	Method
Mode	APMb	ADRc	ADRd	RPMe	Variational with Either $m = 2$ or $m = 5$
${TE}_{4}$	1.461857	1.4618566	1.46186	1.46185	1.461856641
${TE}_{4}$	$- j 7.1558 \times 10^{- 3}$	$- j 7.156 \times 10^{- 3}$	$- j 7.16 \times 10^{- 3}$	$- j 7 \times 10^{- 3}$	$- j 7.155870649 \times 10^{- 3}$
${TE}_{5}$	1.382489	1.3824892	1.38250	1.38239	1.382489223
${TE}_{5}$	$- j 1.8166 \times 10^{- 2}$	$- j 1.8166 \times 10^{- 2}$	$- j 1.817 \times 10^{- 2}$	$- j 1.72 \times 10^{- 2}$	$- j 1.816587736 \times 10^{- 2}$
${TE}_{6}$	1.281364	1.2813644	1.28136	1.28135	1.281364436
${TE}_{6}$	$- j 3.5877 \times 10^{- 2}$	$- j 3.5877 \times 10^{- 2}$	$- j 3.588 \times 10^{- 2}$	$- j 3.30 \times 10^{- 2}$	$- j 3.587739216 \times 10^{- 2}$
${TE}_{7}$	1.142314	1.1423144	1.14231	1.14155	1.142314462
${TE}_{7}$	$- j 5.2876 \times 10^{- 2}$	$- j 5.2876 \times 10^{- 2}$	$- j 5.288 \times 10^{- 2}$	$- j 4.80 \times 10^{- 2}$	$- j 5.287607512 \times 10^{- 2}$
${TE}_{8}$	1.003037	1.0030370	1.00304	1.00304	1.003037019
	$- j 7.0771 \times 10^{- 2}$	$- j 7.0771 \times 10^{- 2}$	$- j 7.077 \times 10^{- 2}$	$- j 6.56 \times 10^{- 1}$	$- j 7.077094109 \times 10^{- 2}$

Abstract

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Equations (15)

Journal of the Optical Society of America B