Wave Manipulation by Topology Optimization - Solid Mechanics

4.1 Transformations Optics 21
a spoon) is manipulated to generate a scattering pattern of a completely different
object (e.g. a cup) and thereby give the illusion of being that object. Unfortunately
no materials in nature posses extreme properties (zero or negative), which makes
realizations challenging. A possible remedy to achieve extreme material properties
is to use metamaterials.
4.1.1 Metamaterials
The concept of metamaterials is to tailor micro structures with a periodicity much
smaller than the wavelength of the incoming wave. The incoming wave does not
”see” the inhomogeneity of the individual micro structure due to the structure being
sub-wavelength. In stead the wave react as if the material is homogenous with
effective material parameters given by the micro structure. This way it is possible to
mix natural occurring materials in a clever way to obtain certain properties which
were otherwise not possible. The idea of metamaterials originates from Veselago
who theoretically studied the effect of both permittivity and permeability being
negative simultaneously[71]. Many years later Pendry proposed the designs of artificial
structured materials which would have effectively negative permittivity and
permeability[72, 73] and not long after the first metamaterial with negative index
of refraction was demonstrated[74]. Shortly after the initial papers on transformation
optics a cloak with approximated extreme anisotropic material properties was
realized in the micro-wave regime using a structured metamaterial[75]. However, realizations
at optical frequencies of the extreme properties even with metamaterials
still remain challenging due to bandwidth-limitations, material absorption and not
to mention fabrication difficulties of sub-wavelength artificial structures.
At a first glance the problem of systematic designing metamaterials using topology
optimization to mitigate some of the challenges seems equivalent to designing
materials with negative Poisson’s ratio[76] and negative thermal expansion[20].
However, obtaining the effective parameters in a systematic and automated way,
which is needed for the topology optimization method to be effective, has proven to
be very challenging. The problem arises from branch ambiguities and that the parameters
vary in a non-trivial way with the angle of incidence[77]. Several methods for
extracting the effective parameters have been reported in the literature[78, 79, 80, 81]
and based on such extraction methods metamaterials with negative permeability
have been designed using topology optimization[82]. However, based on the study
in [82] we do not find it worthwhile at the moment to proceed with double negative
materials. In stead, we take a different path by designing the entire cloaking
structure and thereby bypassing homogenization issues.
4.1.2 All-dielectric cloaking
Several alternative approaches to cloaking have emerged due to all the challenging
problems that arises from extreme material parameters. The most prominent is the