Metasurface Cloaks for Dielectric and
Metallic Elliptical Cylinders and Strips
Hossein M. Bernety and Alexander B. Yakovlev
Department of Electrical Engineering
Center for Applied Electromagnetic Systems Research (CAESR)
University of Mississippi
2014 International Conference on
Electromagnetic in Advanced Applications
ICEAA 2014, Aruba
1
Outline
 Introduction
and Motivation
 Mantle Cloaking
 Formulation
and Theory
 Wave Equation and Mathieu Functions
 Formulation of the Scattering Problem
 Optimum Required Reactance
 Analytical
and Full-wave Simulation Results
 Dielectric Elliptical Cylinders at Microwave and Terahertz Frequencies
 PEC Elliptical Cylinders at Microwave and Terahertz Frequencies
 Strip as a Degenerated Ellipse
 Conclusions
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
2
Cloaking Methods I
Cloak
 Transformation
Optics
 Manipulation of electromagnetic energy flow
Light Rays
 Distorting the ray path by using bulk metamaterials
Object
J. B. Pendry et. al., “Controlling electromagnetic fields,” Science 312, 1780-1782, 2006.
 Plasmonic Method
 Based on scattering cancellation
 Homogeneous and isotropic materials
 Low or negative index materials
 Suppression of the dominant mode
Uncloaked
Cloaked
A. Alu et.al., “Achieving transparency with plasmonic and metamaterial coatings”, Phys. Rev. E. 72, 016623, 2005.
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
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Cloaking Methods II
 Mantle Cloaking
Metasurface Mantle Cloaks
 Based on scattering cancellation
 An ultrathin metasurface
 Anti-phase surface currents
 Suppression of the dominant mode
Electric field distributions
Uncloaked
Cloaked
A. Alu, “Mantle cloak: Invisibility induced by a surface”, Phys. Rev. B. 80, 245115, 2009.
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
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1-D and 2-D Periodic Structures
r
r
w
E
D
E
Mesh Grids
Vertical Strips
PEC
PEC
g
E
E
D
Capacitive Rings
Patch Arrays
Y. R. Padooru et.al., J. Appl. Phys. , vol. 112, pp. 0349075, 2012.
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
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Mantle Cloaking using Graphene
 The thinnest possible mantle cloak
 Large tunability
 Applicable for low terahertz (THz) frequencies
Planar
Structure
Dielectric
Cylinder
P. Y. Chen and A. Alú, “Atomically thin surface cloak using graphene monolayers”, ACS NANO, vol. 5, no. 7, pp.
5855-5863, 2011.
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
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Graphene Nanopatches
 Graphene monolayer provides inductive surface impedance
 A conducting object needs capacitive surface impedance to be cloaked
 To resolve this issue, a patterned graphene metasurface is proposed, which
owns dual capacitive/inductive inductance and can be used to cloak both
dielectric and conducting objects
r
PEC
: surface resistance per unit cell
: surface reactance per unit cell
: periodicity size
: gap size
: relative permittivity of the dielectric cylinder or the spacer
Y. R. Padooru et.al., “Dual capacitive-inductive nature of periodic graphene patches: Transmission characteristics at
low-terahertz frequencies”, Phys. Rev. B, vol. 87, pp. 115401, 2013.
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
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Cloaking using Graphene Nanopatches
Dielectric Cylinder
PEC Cylinder
P. Y. Chen et. al, “Nanostructured graphene metasurface for tunable terahertz cloaking,” New. J. Phys., vol. 15, pp.
123029, 2013.
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
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Elliptical Cloak Designs at Microwaves
r
E
H
r
PEC
E
H
E

H

r
PEC
E
H


c
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
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Elliptical Cloak Designs at THz Frequencies
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
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Scattering from Strips by Mathieu Functions
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
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Elliptical Coordinate System
Focus
Scale factors
: represents an ellipse
: represents a hyperbola
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
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Mathieu Equation
 Two-dimentional Helmholtz Equation:
 where:
 Using the method of separation of variables we have:
Radial Mathieu Equation
Angular Mathieu Equation
 The radial Mathieu equation has four kinds of solution as:
 The angular Mathieu equation has the solution as:
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
p, m can be
even or odd
ICEAA 2014, Aruba
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Formulation of the Scattering Problem
Incident Electric Field
Scattered Electric Field
Incident Magnetic Field
Scattered Magnetic Field
Transmitted Electric Field
Transmitted Magnetic Field
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
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Boundary Conditions
 By applying boundary conditions :
Sheet Impedance Boundary Condition
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
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Matrix Equation for Coefficients
 Now, we apply the orthogonality property of angular Mathieu function as below:
 After some manipulation we come to the matrix equation below:
 Then, unknown coefficients are found
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
16
Bistatic Scattering Width
 To find the scattered field for farfield region, we use the asymptotic form of the radial
Mathieu function and we have:
It has already been found by solving the matrix equation
 The two-dimensional bistatic cross section is defined as:
 Finally, we have:
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
17
Optimum Reactance in terms of Observation Angle
 To find the optimum reactance versus observation angle, we solve the matrix equation for
the dominant scattering mode and require
 Finally the optimum required reactance versus observation angle can be found as:
 With the same approach we can find the optimum required reactance versus observation
angle for the dielectric elliptical cylinder as:
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
18
Optimum Required Reactance
PEC
Dielectric
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
19
Quasi-static Closed-form Condition
 In the quasi-static limit(
), the closed-form condition
for a PEC elliptical cylinder under TM-polarized illumination can be derived as:
 And also, the closed-form condition for a dielectric elliptical cylinder under TM-polarized
illumination can be derived as:
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
20
Surface Reactance Frequency Dispersion
 Frequency dispersion of the surface reactance for graphene monolayer and nanopatches
with respect to the optimum required is found as:
Required Reactance
for Dielectric
Required Reactance
for PEC
Required Reactance
for Strip
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
21
Surface Conductivity of Graphene
Kubo Formula
Intraband Contributions
Interband Contributions
Zs  1 
: charge of electron
: angular frequency
: chemical potential
: temperature
: energy
: reduced Planck’s constant
: momentum relaxation time
G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene”,
J. Appl. Phys., vol. 103, pp. 064302, 2008.
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
22
Dielectric Elliptical Cylinder at THz Frequencies
 Geometry parameters are:
 The required reactance is found to be:
 The design parameters are:
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
23
Power Flow and Far-field Pattern
Uncloaked
Cloaked
f= 3 THz
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
24
Electric Field Distribution
Uncloaked
f= 3 THz
Cloaked
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
25
Closely Spaced and Overlapping Dielectric Elliptical Cylinders
Uncloaked
Cloaked
l= 50 µm
f= 3 THz
Uncloaked
Cloaked
l= 48 µm
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
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Cluster of Dielectric Elliptical Cylinders I
Uncloaked
l= 50 µm= 0.5 λ
Cloaked
g= 3 µm
Uncloaked
Cloaked
f= 3 THz
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
27
Cluster of Dielectric Elliptical Cylinders II
Uncloaked
Cloaked
g= 3 µm
Uncloaked
Cloaked
f= 3 THz
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
28
Overlapping of Dielectric Elliptical Cylinders
Uncloaked
Cloaked
l= 140 µm= 1.4 λ
f= 3 THz
Uncloaked
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
Cloaked
ICEAA 2014, Aruba
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Dielectric Elliptical Cylinder at Microwave Frequencies
 Geometry parameters are:
r
N= 8
E
H

 The required reactance is found to be:
 The design parameters are:
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
30
Dielectric Elliptical Cylinder at Microwave Frequencies
 Geometry parameters are:
r
E
H

 The required reactance is found to be:
 The design parameters are:
N= 3
N= 8
N= 3
N= 8
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
31
Electric Field Distribution
Uncloaked
N= 8
r
f= 3 GHz
Cloaked
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
32
Dielectric Elliptical Cylinder at THz Frequencies
 Geometry parameters are:
 The required reactance is found to be:
 The design parameters are:
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
33
PEC Elliptical Cylinder at THz Frequencies
 Geometry parameters are:
 The required reactance is found to be:
 The design parameters are:
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
34
Electric Field Distribution
Uncloaked
f= 3 THz
Cloaked
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
35
PEC Elliptical Cylinder at Microwave Frequencies
 Geometry parameters are:
PEC
E
H

 The required reactance is found to be:
 The design parameters are:
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
36
2-D Metasurface Cloak for PEC at Microwaves
PEC
E
 Geometry parameters are:
H

 The required reactance is found to be:
 The design parameters are:
N= 10
c
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
37
Power Flow and Far-field Pattern
Uncloaked
Cloaked
c
N= 10
f= 3 GHz
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
38
Electric Field Distribution
Uncloaked
PEC
c
f= 3 GHz
Cloaked
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
39
Strip (Degenerated Ellipse)
 A strip can be modeled as a degenerated ellipse
 Geometry parameters are:
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
40
Power Flow and Far-field Pattern
Uncloaked
Cloaked
f= 3 THz
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
41
Electric Field Distribution
Uncloaked
Cloaked
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
42
Cloaking for Two Strips
Uncloaked
Cloaked
f= 3 THz
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
43
Two Strips Horizontally Oriented
f= 3 THz
Uncloaked
Cloaked
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
44
Two Strips with Overlapping Cloaks
Uncloaked
Cloaked
f= 3 THz
g= 3.7 µm
Uncloaked
Cloaked
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
45
Two Connected Strips
Uncloaked
Cloaked
Uncloaked
f= 3 THz
Cloaked
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
46
Conclusions
An analytical approach has been proposed to cloak elliptical cylinders, and
also, strips at microwave and low-THz frequencies by using conformal mantle
cloak designs.
Although the electromagnetic wave scattering of an elliptical cylinder is
pertinent to the angle of incidence, it is shown that the cloak design is robust
for any incident angle.
The idea has been extended to different geometries such as clusters of
dielectric elliptical cylinders and various cases of strips.
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
47
Conclusions
Challenges for future work???
,
…...
H. M. Bernety and A. B. Yakovlev , “Metasurface Cloaks for
Dielectric and Metallic Elliptical Cylinders and Strips”
ICEAA 2014, Aruba
48