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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Unique determination of periodic polyhedral structures by scattered electromagnetic fields


Authors: Gang Bao, Hai Zhang and Jun Zou
Journal: Trans. Amer. Math. Soc. 363 (2011), 4527-4551
MSC (2010): Primary 35R30, 35Q61, 78A45
Published electronically: April 19, 2011
MathSciNet review: 2806682
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Abstract: This work is concerned with the unique determination of a periodic diffraction grating profile in three dimensions by some scattered electromagnetic fields measured above the grating. In general, it is well known that global uniqueness may not be true when the measurement is only taken for one incident field. Our goal is to completely characterize the global uniqueness properties when the periodic structure is of polyhedral type. Corresponding to each incident plane wave, we are able to classify all unidentifiable structures into three classes and show that any periodic polyhedral structure can be uniquely determined by one incident field if and only if it belongs to none of the three classes. Consequently, the minimum number of incident waves required for the unique determination of a periodic polyhedral structure can be easily read.


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Additional Information

Gang Bao
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027
Email: bao@math.msu.edu

Hai Zhang
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027
Email: zhangh20@msu.edu

Jun Zou
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, People’s Republic of China
Email: zou@math.cuhk.edu.hk

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05334-1
PII: S 0002-9947(2011)05334-1
Keywords: Periodic structure, inverse scattering, uniqueness, dihedral group.
Received by editor(s): April 30, 2009
Published electronically: April 19, 2011
Additional Notes: The first author was supported in part by the NSF grants DMS-0604790, DMS-0908325, CCF-0830161, EAR-0724527, and DMS-0968360, and by the ONR grant N00014-02-1-0365.
The third author was substantially supported by Hong Kong RGC grants (Projects 404606 and 404407) and partially supported by the Cheung Kong Scholars Programme through Wuhan University.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.