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Unique determination of periodic polyhedral structures by scattered electromagnetic fields II: The resonance case


Authors: Gang Bao, Hai Zhang and Jun Zou
Journal: Trans. Amer. Math. Soc. 366 (2014), 1333-1361
MSC (2010): Primary 35R30, 78A46
DOI: https://doi.org/10.1090/S0002-9947-2013-05761-3
Published electronically: October 28, 2013
MathSciNet review: 3145733
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Abstract: This paper is concerned with the unique determination of a three-dimensional polyhedral bi-periodic diffraction grating by the scattered electromagnetic fields measured above the grating. It is shown that the uniqueness by any given incident field fails for seven simple classes of regular polyhedral gratings. Moreover, if a regular bi-periodic polyhedral grating is not uniquely identifiable by a given incident field, then it belongs to a non-empty class of the seven classes whose elements generate the same total field as the original grating when impinged upon by the same incident field. The new theory provides a complete answer to the unique determination of regular bi-periodic polyhedral gratings without any restrictions on Rayleigh frequencies, thus extending our early results (2011) which work under the assumption of no Rayleigh frequencies.


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Additional Information

Gang Bao
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China – and – Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027
Email: drbaogang@gmail.com

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: https://doi.org/10.1090/S0002-9947-2013-05761-3
Keywords: Inverse electromagnetic scattering, periodic diffraction grating, unique determination, dihedral group.
Received by editor(s): January 12, 2011
Received by editor(s) in revised form: November 11, 2011, and November 30, 2011
Published electronically: October 28, 2013
Additional Notes: The research of the first author was supported in part by the NSF grants DMS-0908325, DMS-0968360, and DMS-1211292 and the ONR grants N00014-09-1-0384 and N00014-12-1-0319, a key Project of the Major Research Plan of NSFC (No. 91130004), as well as a special grant from Zhejiang University.
The research of the second author was supported by DMS-0908325.
The research of the third author was substantially supported by Hong Kong RGC grants (Projects 404407 and 405110).
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.