Analysis of diffraction gratings by using an edge element method

Kokou Dossou; Muthukumaran Packirisamy; Marie Fontaine

doi:10.1364/JOSAA.22.000278

Journal of the Optical Society of America A
Vol. 22,
Issue 2,
pp. 278-288
(2005)
•https://doi.org/10.1364/JOSAA.22.000278

Analysis of diffraction gratings by using an edge element method

Kokou Dossou, Muthukumaran Packirisamy, and Marie Fontaine

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Kokou Dossou, Muthukumaran Packirisamy, and Marie Fontaine, "Analysis of diffraction gratings by using an edge element method," J. Opt. Soc. Am. A 22, 278-288 (2005)

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Abstract

Typically the grating problem is formulated for TE and TM polarizations by using, respectively, the electric and magnetic fields aligned with the grating wall and perpendicular to the plane of incidence, and this leads to a one-field-component problem. For some grating profiles such as metallic gratings with a triangular profile, the prediction of TM polarization by using a standard finite-element method experiences a slower convergence rate, and this reduces the accuracy of the computed results and also introduces a numerical polarization effect. This discrepancy cannot be seen as a simple numerical issue, since it has been observed for different types of numerical methods based on the classical formulation. Hence an alternative formulation is proposed, where the grating problem is modeled by taking the electric field as unknown for TM polarization. The application of this idea to both TE and TM polarizations leads to a two-field-component problem. The purpose of the paper is to propose an edge finite-element method to solve this wave problem. A comparison of the results of the proposed formulation and the classical formulation shows improvement and robustness in the new approach.

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Equations (46)

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Node	Basis Functions
1	$v_{1} = L_{1}^{2} (\nabla_{t} L_{2}^{1})$	$v_{2} = L_{1}^{2} (\nabla_{t} L_{3}^{1})$
2	$v_{3} = L_{2}^{2} (\nabla_{t} L_{3}^{1})$	$v_{4} = L_{2}^{2} (\nabla_{t} L_{1}^{1})$
3	$v_{5} = L_{3}^{2} (\nabla_{t} L_{1}^{1})$	$v_{6} = L_{3}^{2} (\nabla_{t} L_{2}^{1})$
4	$v_{7} = L_{4}^{2} (\nabla_{t} (L_{2}^{1} - L_{1}^{1}))$	$v_{8} = L_{4}^{2} (\nabla_{t} L_{3}^{1})$
5	$v_{9} = L_{5}^{2} (\nabla_{t} (L_{3}^{1} - L_{2}^{1}))$	$v_{10} = L_{5}^{2} (\nabla_{t} L_{1}^{1})$
6	$v_{11} = L_{6}^{2} (\nabla_{t} (L_{1}^{1} - L_{3}^{1}))$	$v_{12} = L_{6}^{2} (\nabla_{t} L_{2}^{1})$

Order	C Method	FMM	FEM
Order	C Method	FMM	$v_{z}$	$v_{t}$
TE polarization
$R_{- 2}$	0.5132	0.5134	0.5139	0.5141
$R_{- 1}$	0.1485	0.1476	0.1480	0.1480
$R_{0}$	0.1358	0.1345	0.1350	0.1350
$R_{+ 1}$	0.05891	0.05810	0.05834	0.05834
TM polarization
$R_{- 2}$	0.7004	0.4428	0.6984	0.6996
$R_{- 1}$	0.02767	0.02019	0.02834	0.02809
$R_{0}$	0.02499	0.007737	0.02500	0.02499
$R_{+ 1}$	0.009757	0.0002584	0.01035	0.01011

	FEM
	$v_{z}$			$v_{t}$
Number of triangles	556	2144	8402	556	2144	8402
Number of nodes	1175	4417	17,065	1175	4417	17,065
Truncation ordera	$P = 7$	$P = 15$	$P = 30$	$P = 7$	$P = 15$	$P = 30$
Order
TE polarization
$R_{- 2}$	0.5087	0.5133	0.5139	0.5174	0.5145	0.5141
$R_{- 1}$	0.1480	0.1479	0.1480	0.1482	0.1481	0.1480
$R_{0}$	0.1343	0.1349	0.1350	0.1354	0.1350	0.1350
$R_{+ 1}$	0.05870	0.05837	0.05834	0.05814	0.05831	0.05834
CPU (s)	0.54	1.64	19.69	2.03	9.86	130.53
TM polarization
$R_{- 2}$	0.6973	0.6983	0.6984	0.7003	0.6999	0.6996
$R_{- 1}$	0.03138	0.02873	0.02834	0.02639	0.02783	0.02809
$R_{0}$	0.02680	0.02520	0.02500	0.02437	0.02496	0.02499
$R_{+ 1}$	0.01239	0.01062	0.01035	0.009111	0.009942	0.01011
CPU (s)	0.56	1.65	19.72	2.28	9.86	133.71

Mesh	NE	NN	$P_{1}$	$P_{2}$
(1)	2763	5666	-32	10
(2)	4806	9799	-36	14
(3)	10,919	22,122	-36	14
(4)	30,313	61,102	-82	60

Node	Basis Functions
1	$v_{1} = L_{1}^{2} (\nabla_{t} L_{2}^{1})$	$v_{2} = L_{1}^{2} (\nabla_{t} L_{3}^{1})$
2	$v_{3} = L_{2}^{2} (\nabla_{t} L_{3}^{1})$	$v_{4} = L_{2}^{2} (\nabla_{t} L_{1}^{1})$
3	$v_{5} = L_{3}^{2} (\nabla_{t} L_{1}^{1})$	$v_{6} = L_{3}^{2} (\nabla_{t} L_{2}^{1})$
4	$v_{7} = L_{4}^{2} (\nabla_{t} (L_{2}^{1} - L_{1}^{1}))$	$v_{8} = L_{4}^{2} (\nabla_{t} L_{3}^{1})$
5	$v_{9} = L_{5}^{2} (\nabla_{t} (L_{3}^{1} - L_{2}^{1}))$	$v_{10} = L_{5}^{2} (\nabla_{t} L_{1}^{1})$
6	$v_{11} = L_{6}^{2} (\nabla_{t} (L_{1}^{1} - L_{3}^{1}))$	$v_{12} = L_{6}^{2} (\nabla_{t} L_{2}^{1})$

Abstract

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Figures (9)

Tables (4)

Equations (46)

Journal of the Optical Society of America A