An adaptive boundary element for eigenvalue problems with the Helmholtz equation: simplified h-scheme

1994 ◽  
Vol 19 (3) ◽  
pp. 143-147 ◽  
Author(s):  
N. Kamiya ◽  
K. Nogae ◽  
S.-Q. Xu
Keyword(s):  
Helmholtz Equation ◽  
Boundary Element ◽  
Eigenvalue Problems

scholarly journals Whispering Bloch modes

2016 ◽  
Vol 472 (2191) ◽  
pp. 20160103 ◽  
Author(s):  
B. Maling ◽  
R. V. Craster
Keyword(s):  
Helmholtz Equation ◽  
Eigenvalue Problems ◽  
Open Systems ◽  
Rotational Symmetry ◽  
Field Enhancement ◽  
High Order ◽  
Whispering Gallery Modes ◽  
Whispering Gallery ◽  
Field Patterns ◽  
Bloch Modes

We investigate eigenvalue problems for the planar Helmholtz equation in open systems with a high order of rotational symmetry. The resulting solutions have similarities with the whispering gallery modes exploited in photonic micro-resonators and elsewhere, but unlike these do not necessarily require a surrounding material boundary, with confinement instead resulting from the geometry of a series of inclusions arranged in a ring. The corresponding fields exhibit angular quasi-periodicity reminiscent of Bloch waves, and hence we refer to them as whispering Bloch modes (WBMs). We show that if the geometry of the system is slightly perturbed such that the rotational symmetry is broken, modes with asymmetric field patterns can be observed, resulting in field enhancement and other potentially desirable effects. We investigate the WBMs of two specific geometries first using expansion methods and then by applying a two-scale asymptotic scheme.

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