Reciprocity Theorems and Their Application to Numerical Analysis in Grating Theory

Junichi NAKAYAMA; Yasuhiko TAMURA

doi:10.1587/transele.E100.C.3

Reciprocity Theorems and Their Application to Numerical Analysis in Grating Theory

Junichi NAKAYAMA, Yasuhiko TAMURA

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This paper deals with the diffraction of a monochromatic plane wave by a periodic grating. We discuss a problem how to obtain a numerical diffraction efficiency (NDE) satisfying the reciprocity theorem for diffraction efficiencies, because diffraction efficiencies are the subject of the diffraction theories. First, this paper introduces a new formula that decomposes an NDE into two components: the even component and the odd one. The former satisfies the reciprocity theorem for diffraction efficiencies, but the latter does not. Therefore, the even component of an NDE becomes an answer to our problem. On the other hand, the odd component of an NDE represents an unwanted error. Using such the decomposition formula, we then obtain another new formula that decomposes the conventional energy error into two components. One is the energy error made by even components of NDE's. The other is the energy error constructed by unwanted odd ones and it may be used as a reciprocity criterion of a numerical solution. This decomposition formula shows a drawback of the conventional energy balance. The total energy error is newly introduced as a more strict condition for a desirable solution. We point out theoretically that the reciprocal wave solution, an approximate solution satisfying the reciprocity for wave fields, gives another solution to our problem. Numerical examples are given for the diffraction of a TM plane wave by a very rough periodic surface with perfect conductivity. In the case of a numerical solution by the image integral equation of the second kind, we found that the energy error is much reduced by use of the even component of an NDE as an approximate diffraction efficiency or by use of a reciprocal wave solution.

Publication: IEICE TRANSACTIONS on Electronics Vol.E100-C No.1 pp.3-10

Publication Date: 2017/01/01

Publicized

Online ISSN: 1745-1353

DOI: 10.1587/transele.E100.C.3

Type of Manuscript: Special Section PAPER (Special Section on Recent Progress in Electromagnetic Theory and Its Application)

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Authors

Junichi NAKAYAMA
Kyoto Institute of Technology
Yasuhiko TAMURA
Kyoto Institute of Technology

Keyword

wave diffraction, periodic surface, reciprocity theorem, decomposition of diffraction efficiencies, decomposition of energy errors, image integral equation

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Junichi NAKAYAMA, Yasuhiko TAMURA, "Reciprocity Theorems and Their Application to Numerical Analysis in Grating Theory" in IEICE TRANSACTIONS on Electronics, vol. E100-C, no. 1, pp. 3-10, January 2017, doi: 10.1587/transele.E100.C.3.
Abstract: This paper deals with the diffraction of a monochromatic plane wave by a periodic grating. We discuss a problem how to obtain a numerical diffraction efficiency (NDE) satisfying the reciprocity theorem for diffraction efficiencies, because diffraction efficiencies are the subject of the diffraction theories. First, this paper introduces a new formula that decomposes an NDE into two components: the even component and the odd one. The former satisfies the reciprocity theorem for diffraction efficiencies, but the latter does not. Therefore, the even component of an NDE becomes an answer to our problem. On the other hand, the odd component of an NDE represents an unwanted error. Using such the decomposition formula, we then obtain another new formula that decomposes the conventional energy error into two components. One is the energy error made by even components of NDE's. The other is the energy error constructed by unwanted odd ones and it may be used as a reciprocity criterion of a numerical solution. This decomposition formula shows a drawback of the conventional energy balance. The total energy error is newly introduced as a more strict condition for a desirable solution. We point out theoretically that the reciprocal wave solution, an approximate solution satisfying the reciprocity for wave fields, gives another solution to our problem. Numerical examples are given for the diffraction of a TM plane wave by a very rough periodic surface with perfect conductivity. In the case of a numerical solution by the image integral equation of the second kind, we found that the energy error is much reduced by use of the even component of an NDE as an approximate diffraction efficiency or by use of a reciprocal wave solution.
URL: https://global.ieice.org/en_transactions/electronics/10.1587/transele.E100.C.3/_p

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@ARTICLE{e100-c_1_3,
author={Junichi NAKAYAMA, Yasuhiko TAMURA, },
journal={IEICE TRANSACTIONS on Electronics},
title={Reciprocity Theorems and Their Application to Numerical Analysis in Grating Theory},
year={2017},
volume={E100-C},
number={1},
pages={3-10},
abstract={This paper deals with the diffraction of a monochromatic plane wave by a periodic grating. We discuss a problem how to obtain a numerical diffraction efficiency (NDE) satisfying the reciprocity theorem for diffraction efficiencies, because diffraction efficiencies are the subject of the diffraction theories. First, this paper introduces a new formula that decomposes an NDE into two components: the even component and the odd one. The former satisfies the reciprocity theorem for diffraction efficiencies, but the latter does not. Therefore, the even component of an NDE becomes an answer to our problem. On the other hand, the odd component of an NDE represents an unwanted error. Using such the decomposition formula, we then obtain another new formula that decomposes the conventional energy error into two components. One is the energy error made by even components of NDE's. The other is the energy error constructed by unwanted odd ones and it may be used as a reciprocity criterion of a numerical solution. This decomposition formula shows a drawback of the conventional energy balance. The total energy error is newly introduced as a more strict condition for a desirable solution. We point out theoretically that the reciprocal wave solution, an approximate solution satisfying the reciprocity for wave fields, gives another solution to our problem. Numerical examples are given for the diffraction of a TM plane wave by a very rough periodic surface with perfect conductivity. In the case of a numerical solution by the image integral equation of the second kind, we found that the energy error is much reduced by use of the even component of an NDE as an approximate diffraction efficiency or by use of a reciprocal wave solution.},
keywords={},
doi={10.1587/transele.E100.C.3},
ISSN={1745-1353},
month={January},}

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TY - JOUR
TI - Reciprocity Theorems and Their Application to Numerical Analysis in Grating Theory
T2 - IEICE TRANSACTIONS on Electronics
SP - 3
EP - 10
AU - Junichi NAKAYAMA
AU - Yasuhiko TAMURA
PY - 2017
DO - 10.1587/transele.E100.C.3
JO - IEICE TRANSACTIONS on Electronics
SN - 1745-1353
VL - E100-C
IS - 1
JA - IEICE TRANSACTIONS on Electronics
Y1 - January 2017
AB - This paper deals with the diffraction of a monochromatic plane wave by a periodic grating. We discuss a problem how to obtain a numerical diffraction efficiency (NDE) satisfying the reciprocity theorem for diffraction efficiencies, because diffraction efficiencies are the subject of the diffraction theories. First, this paper introduces a new formula that decomposes an NDE into two components: the even component and the odd one. The former satisfies the reciprocity theorem for diffraction efficiencies, but the latter does not. Therefore, the even component of an NDE becomes an answer to our problem. On the other hand, the odd component of an NDE represents an unwanted error. Using such the decomposition formula, we then obtain another new formula that decomposes the conventional energy error into two components. One is the energy error made by even components of NDE's. The other is the energy error constructed by unwanted odd ones and it may be used as a reciprocity criterion of a numerical solution. This decomposition formula shows a drawback of the conventional energy balance. The total energy error is newly introduced as a more strict condition for a desirable solution. We point out theoretically that the reciprocal wave solution, an approximate solution satisfying the reciprocity for wave fields, gives another solution to our problem. Numerical examples are given for the diffraction of a TM plane wave by a very rough periodic surface with perfect conductivity. In the case of a numerical solution by the image integral equation of the second kind, we found that the energy error is much reduced by use of the even component of an NDE as an approximate diffraction efficiency or by use of a reciprocal wave solution.
ER -