Solution of the vector wave equation by the separable effective adiabatic basis set method

Kirill Gokhberg; Ilya Vorobeichik; Edvardas Narevicius; Nimrod Moiseyev

doi:10.1364/JOSAB.21.001809

Journal of the Optical Society of America B
Vol. 21,
Issue 10,
pp. 1809-1817
(2004)
•https://doi.org/10.1364/JOSAB.21.001809

Solution of the vector wave equation by the separable effective adiabatic basis set method

Kirill Gokhberg, Ilya Vorobeichik, Edvardas Narevicius, and Nimrod Moiseyev

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Kirill Gokhberg, Ilya Vorobeichik, Edvardas Narevicius, and Nimrod Moiseyev, "Solution of the vector wave equation by the separable effective adiabatic basis set method," J. Opt. Soc. Am. B 21, 1809-1817 (2004)

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Abstract

A novel separable effective adiabatic basis (SEAB) for the solution of the transverse vector wave equation by the variational method is presented. The basis is constructed by a suitably modified adiabatic approximation. The method of SEAB construction is applicable to the waveguides of a general cross section. By calculating scalar modes in rectangular and rib waveguides, we show that the use of SEAB entails computational effort several orders of magnitude less than the use of the more conventional Fourier basis. As an illustrative example, the polarized modes of a rib waveguide are calculated.

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Number of Basis Functions, $ϕ_{l} = X_{i} (x) Y_{j} (y)$	Sine Basis		Adiabatic Basis
Number of Basis Functions, $ϕ_{l} = X_{i} (x) Y_{j} (y)$	$n_{eff}$	Error	$n_{eff}$	Error
$i_{max} = 40,$ $j_{max} = 10,$ $l_{max} = 400$	1.453523	—	1.457050	—
$i_{max} = 60,$ $j_{max} = 15,$ $l_{max} = 900$	1.455190	$1.667 \times 10^{- 3}$	1.457102	$5.2 \times 10^{- 5}$
$i_{max} = 80,$ $j_{max} = 20,$ $l_{max} = 1600$	1.456160	$9.70 \times 10^{- 4}$	1.457141	$3.9 \times 10^{- 5}$
$i_{max} = 100,$ $j_{max} = 25,$ $l_{max} = 2500$	1.456459	$2.99 \times 10^{- 4}$	1.457152	$1.1 \times 10^{- 5}$

Number of Basis Functions, $ϕ_{l} = X_{i} (x) Y_{j} (y)$	Sine Basis		Adiabatic Basis
Number of Basis Functions, $ϕ_{l} = X_{i} (x) Y_{j} (y)$	$n_{eff}$	Error	$n_{eff}$	Error
$i_{max} = 40,$ $j_{max} = 10,$ $l_{max} = 400$	1.471733	—	1.473935	—
$i_{max} = 60,$ $j_{max} = 15,$ $l_{max} = 900$	1.473420	$1.687 \times 10^{- 3}$	1.473947	$1.2 \times 10^{- 5}$
$i_{max} = 80,$ $j_{max} = 20,$ $l_{max} = 1600$	1.473587	$1.67 \times 10^{- 4}$	1.473950	$3 \times 10^{- 6}$
$i_{max} = 100,$ $j_{max} = 25,$ $l_{max} = 2500$	1.473691	$1.04 \times 10^{- 4}$	1.473952	$2 \times 10^{- 6}$

Number of Basis Functions, $ϕ_{l} = X_{i} (x) Y_{j} (y)$	Sine Basis		Adiabatic Basis
Number of Basis Functions, $ϕ_{l} = X_{i} (x) Y_{j} (y)$	$n_{eff}$	Error	$n_{eff}$	Error
$i_{max} = 40,$ $j_{max} = 10,$ $l_{max} = 400$	1.444219	—	1.445291	—
$i_{max} = 60,$ $j_{max} = 15,$ $l_{max} = 900$	1.444364	$1.45 \times 10^{- 4}$	1.445351	$6.0 \times 10^{- 5}$
$i_{max} = 80,$ $j_{max} = 20,$ $l_{max} = 1600$	1.444693	$3.29 \times 10^{- 4}$	1.445376	$2.5 \times 10^{- 5}$
$i_{max} = 100,$ $j_{max} = 25,$ $l_{max} = 2500$	1.444741	$4.8 \times 10^{- 5}$	1.445386	$1.0 \times 10^{- 5}$

Number of Basis Functions, $ϕ_{l} = X_{i} (x) Y_{j} (y)$	TM Mode		TE Mode
Number of Basis Functions, $ϕ_{l} = X_{i} (x) Y_{j} (y)$	$n_{eff}$	Error	$n_{eff}$	Error
$i_{max} = 40,$ $j_{max} = 10,$ $l_{max} = 400$	1.457153	—	1.457942	—
$i_{max} = 60,$ $j_{max} = 15,$ $l_{max} = 900$	1.457205	$5.2 \times 10^{- 5}$	1.458009	$6.7 \times 10^{- 5}$
$i_{max} = 80,$ $j_{max} = 20,$ $l_{max} = 1600$	1.457249	$4.4 \times 10^{- 5}$	1.458057	$4.8 \times 10^{- 5}$

Number of Basis Functions, $ϕ_{l} = X_{i} (x) Y_{j} (y)$	TM Mode		TE Mode
Number of Basis Functions, $ϕ_{l} = X_{i} (x) Y_{j} (y)$	$n_{eff}$	Error	$n_{eff}$	Error
$i_{max} = 40,$ $j_{max} = 10,$ $l_{max} = 400$	1.473997	—	1.474862	—
$i_{max} = 60,$ $j_{max} = 15,$ $l_{max} = 900$	1.474012	$1.5 \times 10^{- 5}$	1.474872	$1.0 \times 10^{- 5}$
$i_{max} = 80,$ $j_{max} = 20,$ $l_{max} = 1600$	1.474016	$4 \times 10^{- 6}$	1.474875	$3 \times 10^{- 6}$

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