Metamaterial-Based Cylinders Used for Invisible Cloak Realization – AFRL-AFOSR-UK-TR-2011-0064

Posted on April 20, 2015


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The realization of structures that do not scatter electromagnetic field, i.e. structures that
appear invisible for EM waves in outer space, is not a new concept. The possibility of a plane
wave passing without distortions through a structure with anisotropic filling was theoretically
first investigated in 1960s, see [1]. The basis of the work was the invariance property of
Maxwell’s equations with respect to transformation of space metric and permeability and
permittivity tensors of the medium. In [2]-[5] it was shown that for certain combinations of
permittivities in a two-layer dielectric ellipsoid, the scattered field is zero. The analysis
revealed that at least one of the layers should have relative permittivity smaller than one (i.e.
local negative polarizability, which is inherent to plasmonic materials and plasma-like ENG
metamaterials), and moreover such structures have the advantage of not requiring anisotropic
materials. In [6], hard surfaces were used in the design of supporting struts of reflector
antenna feeds, to reduce their blockage in reflector systems. In this way it is possible to
considerably reduce the scattered field for one angle of incidence, for both polarizations of
the incident wave. Several other concepts for obtaining invisible scatterers were proposed,
like minimum scattering antennas and active scatterers ([7], [8]).
The possibility of cloaking objects using a metamaterial cover has extensively been
studied (see e.g. [9]-[13]). Metamaterials are new composite materials that recently have
found applications in the design of antennas and microwave components. These special
materials can be formed by periodic arrangements of many sub-wavelength inclusions in a
dielectric environment, in such a way as to achieve macroscopic electromagnetic or optical
properties that cannot be found in nature. In the metamaterial cloak approach, material has
been used to render a volume effectively invisible to incident radiation, i.e. to squeeze space
from a volume into a shell surrounding the concealment volume. The perfect cloak ensures
that for any incident field the electromagnetic scattered field vanishes in the free-space
external to the cloaking shell, and the total field vanishes inside the free-space cavity of the
shell. Thus, any object placed in the cavity does not perturb the electromagnetic field outside
the cloak, and an external observer does not see the object and the cloak, like they are absent.
However, the coordinate transformations that are used for cloak design do not affect the form
of Maxwell’s equations, but they influence permittivity and permeability tensors.
Consequently, the materials needed for building the cloak are fully anisotropic with spatiallyvarying
constitutive parameters.

The first practical realization of a cloak was made by Schurig et al. in 2006 [14]. The
realized cloak was not fully anisotropic, but it was designed to work for only one polarization
(TMz polarization). By this design of metamaterial layers with only the radial dependence of
the radial permeability component was needed, and it was realized using the split-ring
resonators. The described cloak was prototyped and embedded into scattering chamber. The
distribution of the vertical component of the electric field around the cloak was scanned, and
the obtained results showed some decrease of the scattered field. Although this experiment
proved the basic idea of cloaking, the level of scattered field has not been quantified.
Previous work in invisibility, therefore, gives rise to several questions:
1. Why the developed cloak with the reduced variation of constitutive parameters (i.e.
cloaks made from uniaxial materials) scatters the electromagnetic field, even in the
ideal case?
2. Is it possible to realize a cloak with the reduced variation of constitutive parameters
that is entirely invisible (at least at the central frequency)?
3. What is the bandwidth of a cloak that is built using metamaterial layers? Is it possible
to enlarge the bandwidth of such a cloak?
4. Is it possible to realize a cloak with the reduced variation of constitutive parameters
that work for oblique direction of incident wave?
5. Is it possible to realize a cloak that is invisible to pulse excitation (i.e. to radar)?
6. Is it possible to extend the single-curved models (cloaks) to double-curved structures?
The purpose of the project is, therefore, to provide further analysis of this issue.
The possible application of the developed cloak will be reduction of the scattered field
from some mechanical structure. In more details, there are many situations where
electromagnetic waves are obstructed by some mechanical structure. The obstruction may
represent aperture blockage causing increased sidelobes and reduced gain, if the structure is a
part of the antenna system or if it is placed close to the antenna. Examples of such structures
are objects placed in the vicinity of the antenna system (e.g. musts on ships near radars or
communication systems), supporting struts in large reflector systems, etc. In all such systems
the frequency, polarization and angle of incidence are known which allows us to construct a
special structure that will reduce the scattered field.

 

TABLE OF CONTENTS
1 INTRODUCTION…………………………………………………………………………………………………………………………. 4
2 PROJECT OBJECTIVE AND REALIZED OUTCOMES ……………………………………………………………… 7
3 ANALYSIS OF UNIAXIAL MULTILAYER CYLINDERS USED FOR INVISIBLE CLOAK
REALIZATION……………………………………………………………………………………………………………………………….. 10
3.1. INTRODUCTION ………………………………………………………………………………………………………………………………11
3.2. THEORETICAL CONSIDERATIONS………………………………………………………………………………………………….13
3.3. RESULTS OF THE CLOAK ANALYSIS ………………………………………………………………………………………………14
3.3.1. Ideal cloak ………………………………………………………………………………………………………………………. 15
3.3.2. TMz cloak (Schurig cloak) …………………………………………………………………………………………………. 19
3.3.3. TEz cloak (Cai cloak)………………………………………………………………………………………………………… 21
3.4. LIMITATIONS OF SIMPLIFIED CLOAK REALIZATIONS…………………………………………………………………..22
4 OPTIMIZATION OF UNIAXIAL MULTILAYER CYLINDERS USED FOR INVISIBLE CLOAK
REALIZATION……………………………………………………………………………………………………………………………….. 29
4.1 GLOBAL OPTIMIZATION TECHNIQUES………………………………………………………………………………………………………..30
4.1.1. Classical particle swarm optimization …………………………………………………………………………………. 30
4.1.2. Particle swarm optimization with local best topology ……………………………………………………………. 36
4.1.3. Comprehensive learning particle swarm optimization……………………………………………………………. 37
4.1.4. Performance comparison …………………………………………………………………………………………………… 39
4.2 OPTIMIZED CLOAK ………………………………………………………………………………………………………………………………..41
4.2.1. Number of layers ………………………………………………………………………………………………………………. 43
4.2.2. Losses and tolerance …………………………………………………………………………………………………………. 45
4.2.3. Bandwidth – optimized cloak ………………………………………………………………………………………………. 47
4.2.4. Three layers cloak …………………………………………………………………………………………………………….. 48
5 OBLIQUE INCIDENCE PLANE WAVE SCATTERING BY SCHURIG CLOAK …………………………. 52
5.1. ANALYSISMETHOD………………………………………………………………………………………………………………………..53
5.2. RESULTS ………………………………………………………………………………………………………………………………………….67
6 ANALYSIS OF UNIAXIAL MULTILAYER SPHERICAL STRUCTURES USED FOR INVISIBLE
CLOAK REALIZATION………………………………………………………………………………………………………………….. 72
6.1. INTRODUCTION ………………………………………………………………………………………………………………………………73
6.2. THEORETICAL CONSIDERATIONS………………………………………………………………………………………………….74
6.2.1. Vector eigenvectors …………………………………………………………………………………………………………… 74
6.2.2. Spherical harmonics………………………………………………………………………………………………………….. 76
6.2.3. Modifications for anisotropic structures ………………………………………………………………………………. 79
6.2. RESULTS OF THE CLOAK ANALYSIS ………………………………………………………………………………………………82
6.2.1. Pendry cloak design ………………………………………………………………………………………………………….. 82
6.2.2. Engheta cloak…………………………………………………………………………………………………………………… 83
6.2.3. Optimized cloak………………………………………………………………………………………………………………… 84
CONCLUSIONS………………………………………………………………………………………………………………………………. 86
BIBLIOGRAPHY…………………………………………………………………………………………………………………………….. 89

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