Reduction of computation time for crossed-grating problems: a group-theoretic approach

Benfeng Bai; Lifeng Li

doi:10.1364/JOSAA.21.001886

Journal of the Optical Society of America A
Vol. 21,
Issue 10,
pp. 1886-1894
(2004)
•https://doi.org/10.1364/JOSAA.21.001886

Reduction of computation time for crossed-grating problems: a group-theoretic approach

Benfeng Bai and Lifeng Li

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Benfeng Bai and Lifeng Li, "Reduction of computation time for crossed-grating problems: a group-theoretic approach," J. Opt. Soc. Am. A 21, 1886-1894 (2004)

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Abstract

A systematic approach based on group theory is established to deal with diffraction problems of crossed gratings by exploiting symmetries. With this approach, a problem in an asymmetrical incident mounting can be decomposed into a superposition of several symmetrical basis problems so that the computation efficiency is improved effectively. This methodology offers a convenient and succinct way to treat all possible symmetry cases by following only several mechanical steps instead of intricate mathematical considerations or physical intuition. It is also general, applicable to both scalar-wave and vector-wave problems and in principle can be easily adapted to any numerical method. A numerical example is presented to show its application and effectiveness.

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Plane Point Group		Symmetry Operations
Herman–Manquin Symbol	Schoenflies Symbol	Symmetry Operations
1	$C_{1}$	e
m	$C_{s}$	e , $σ_{v}$
2	$C_{2}$	e , $c_{2}$
$2 mm$	$C_{2 v}$	e , $c_{2},$ $σ_{v},$ $σ_{d}$
3	$C_{3}$	e , $c_{3}^{1},$ $c_{3}^{2}$
$3 m$	$C_{3 v}$	e , $c_{3}^{1},$ $c_{3}^{2},$ $σ_{b},$ $σ_{d},$ $σ_{f}$
4	$C_{4}$	e , $c_{4}^{1},$ $c_{4}^{3},$ $c_{2}$
$4 mm$	$C_{4 v}$	e , $c_{4}^{1},$ $c_{4}^{3},$ $c_{2},$ $σ_{vx},$ $σ_{vy},$ $σ_{da},$ $σ_{db}$
6	$C_{6}$	e , $c_{6}^{1},$ $c_{6}^{5},$ $c_{3}^{1},$ $c_{3}^{2},$ $c_{2}$
$6 mm$	$C_{6 v}$	e , $c_{6}^{1},$ $c_{6}^{5},$ $c_{3}^{1},$ $c_{3}^{2},$ $c_{2},$ $σ_{vx},$ $σ_{vA},$ $σ_{vB},$ $σ_{dy},$ $σ_{dC},$ $σ_{dD}$

Lattice	Point Group	Plane Group^b
$mp$	1	$p 1$
	2	$p 2$
		$pm$
	m	$pg$
		$cm$
$op$ or $oc$
		$p 2 mm$
	$2 mm$	$p 2 mg$
		$p 2 gg$
		$c 2 mm$

tp	4 4mm	p 4 $p 4 mm$ $p 4 gm$

	3	$p 3$
hp	$3 m$	$p 3 m 1$ $p 31 m$
	6	$p 6$
	$6 mm$	$p 6 mm$

IR	e	$c_{2}$	$σ_{x}$	$σ_{y}$
$T^{(1)}$	1	1	1	1
$T^{(2)}$	1	1	-1	-1
$T^{(3)}$	1	-1	-1	1
$T^{(4)}$	1	-1	1	-1

Plane Point Group		Symmetry Operations
Herman–Manquin Symbol	Schoenflies Symbol	Symmetry Operations
1	$C_{1}$	e
m	$C_{s}$	e , $σ_{v}$
2	$C_{2}$	e , $c_{2}$
$2 mm$	$C_{2 v}$	e , $c_{2},$ $σ_{v},$ $σ_{d}$
3	$C_{3}$	e , $c_{3}^{1},$ $c_{3}^{2}$
$3 m$	$C_{3 v}$	e , $c_{3}^{1},$ $c_{3}^{2},$ $σ_{b},$ $σ_{d},$ $σ_{f}$
4	$C_{4}$	e , $c_{4}^{1},$ $c_{4}^{3},$ $c_{2}$
$4 mm$	$C_{4 v}$	e , $c_{4}^{1},$ $c_{4}^{3},$ $c_{2},$ $σ_{vx},$ $σ_{vy},$ $σ_{da},$ $σ_{db}$
6	$C_{6}$	e , $c_{6}^{1},$ $c_{6}^{5},$ $c_{3}^{1},$ $c_{3}^{2},$ $c_{2}$
$6 mm$	$C_{6 v}$	e , $c_{6}^{1},$ $c_{6}^{5},$ $c_{3}^{1},$ $c_{3}^{2},$ $c_{2},$ $σ_{vx},$ $σ_{vA},$ $σ_{vB},$ $σ_{dy},$ $σ_{dC},$ $σ_{dD}$

Lattice	Point Group	Plane Group^b
$mp$	1	$p 1$
	2	$p 2$
		$pm$
	m	$pg$
		$cm$
$op$ or $oc$
		$p 2 mm$
	$2 mm$	$p 2 mg$
		$p 2 gg$
		$c 2 mm$

tp	4 4mm	p 4 $p 4 mm$ $p 4 gm$

	3	$p 3$
hp	$3 m$	$p 3 m 1$ $p 31 m$
	6	$p 6$
	$6 mm$	$p 6 mm$

Abstract

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Figures (9)

Tables (3)

Equations (25)

Journal of the Optical Society of America A