Characterization of negative refraction with multilayered mushroom-type metamaterials at microwaves

Abstract

In this paper, it is shown that bulk metamaterials formed by multilayered mushroom-type structures enable broadband negative refraction. The metamaterial configurations are modeled using homogenization methods developed for a uniaxial wire medium loaded with periodic metallic elements (for example, patch arrays). It is shown that the phase of the transmission coefficient decreases with the increasing incidence angle, resulting in the negative spatial shift of the transmitted wave. The homogenization model results are obtained with the uniform plane-wave incidence, and the full-wave results are generated with a Gaussian beam excitation, showing a strong negative refraction in a significant frequency band. Having in mind a possible experimental verification of our findings, we investigate the effect of introducing air gaps in between the metamaterial layers, showing that even in such simple configuration the negative refraction phenomenon is quite robust.

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Characterization of negative refraction with multilayered mushroom-type
metamaterials at microwaves
Chandra S. R. Kaipa,1,a)Alexander B. Yakovlev,1and Ma ´rio G. Silveirinha2
1Department of Electrical Engineering, The University of Mississippi, University, Mississippi 38677-1848, USA
2Departamento de Engenharia Electrote ´cnica, Instituto de Telecomunicac ¸o ˜es, Universidade de Coimbra Po ´lo II,
3030-290 Coimbra, Portugal
(Received 22 September 2010; accepted 13 December 2010; published online 23 February 2011)
In this paper, it is shown that bulk metamaterials formed by multilayered mushroom-type
structures enable broadband negative refraction. The metamaterial configurations are modeled
using homogenization methods developed for a uniaxial wire medium loaded with periodic
metallic elements (for example, patch arrays). It is shown that the phase of the transmission
coefficient decreases with the increasing incidence angle, resulting in the negative spatial shift of
the transmitted wave. The homogenization model results are obtained with the uniform plane-wave
incidence, and the full-wave results are generated with a Gaussian beam excitation, showing a
strong negative refraction in a significant frequency band. Having in mind a possible experimental
verification of our findings, we investigate the effect of introducing air gaps in between the
metamaterial layers, showing that even in such simple configuration the negative refraction
phenomenon is quite robust. V
C 2011 American Institute of Physics. [doi:10.1063/1.3549129]
I. INTRODUCTION
Negative-index metamaterials have been the subject of
interest in recent years, due to their extraordinary properties
such as, partial focusing, sub-wavelength imaging, and nega-
tive refraction. In particular, the phenomenon of negative
refraction has attracted attention both in the optical and
microwave communities. This phenomenon can in general
be observed in materials with simultaneously negative per-
mittivity and permeability, as originally suggested by Vese-
lago.1However, the emergence of negative refraction due to
a negative phase velocity has been reported much earlier by
Schuster2and Mandelshtam.3Although, negative refraction
is not observed in conventional dielectrics, the advent of
metamaterials brought new opportunities to observe this phe-
nomenon, as reported recently in the literature.
In Refs. 4–6, negative refraction has been realized using
the materials with indefinite anisotropic properties, in which
not all the principal components of the permittivity and per-
meability tensors have the same sign. Also, some other pos-
sibilities include the use of a nonlocal material formed by a
crossed wire mesh, which results in broadband negative
refraction,7and by engineering the dispersion of the photonic
crystals.8Recently, negative refraction was also observed at
optical frequencies by using an array of metallic nano-
rods.9,10However, the design considered in Refs. 9 and 10 is
effective only at optical frequencies. At lower THz and
microwave frequencies the array of nanorods is character-
ized by strong spatial dispersion,11and it behaves very dif-
ferently from a material with indefinite properties.
However, it has been recently shown that the spatial dis-
persion (SD) effects in wire medium, formed by a two-
dimensional lattice of parallel conducting wires, can be sig-
nificantly reduced.12–14In Ref. 12, it was suggested coating
the wires with a magnetic material or attaching large con-
ducting plates to the wires. In Refs. 13 and 14, it has been
shown based on nonlocal and local homogenization models
that the periodic metallic vias in the mushroom structure can
be treated as a uniaxial continuous Epsilon-Negative (ENG)
material loaded with a capacitive grid of patches, with a
proper choice of the period and the thickness of the vias.
Based on these findings, it has been shown in Refs. 15 and
16 that by periodically attaching metallic patches to an array
of parallel wires (with the unit cell resembling in part the
mushroom structure) it is possible to synthesize a multilay-
ered local uniaxial ENG material at longer wavelengths
loaded with patch arrays.
In this work, as a continuation of our preliminary study
in Refs. 15 and 16, we show that by periodically attaching
metallic patches to an array of metallic wires (when SD
effects are significantly reduced) it is possible to mimic the
observed phenomenon of negative refraction from an array
of metallic nanorods at optical frequencies, in the microwave
regime. We present a complete parametric study of the nega-
tive refraction effect, highlighting its dependence on fre-
quency, thickness of the metamaterial slab, and show how it
can be conveniently modeled using effective medium theory.
In addition, we investigate the effect of introducing air gaps
in between the different metamaterial layers [formed by peri-
odically attaching pairs of metallic patches to an array of me-
tallic vias embedded in a single dielectric slab] aiming at a
possible experimental verification of our findings, and show
that such simple configuration enables the control of the neg-
ative refraction angle. The propagation characteristics in the
proposed multilayered mushroom structures are analyzed
using the nonlocal and local homogenization models for the
wire medium (WM). Our results show that there is an excel-
lent agreement between the two homogenization models
over a wide frequency range, which demonstrates, indeed,
a)Author to whom correspondence should be addressed. Electronic mail:
ckaipa@olemiss.edu.
0021-8979/2011/109(4)/044901/10/$30.00
V
C 2011 American Institute of Physics109, 044901-1
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the suppression of SD effects in the WM. The numerical
results are presented for several configurations (with and
without the air gaps) showing a broadband strong negative
refraction at microwave frequencies.
The paper is organized as follows. In Sec. II, the formal-
ism of nonlocal and local homogenization models is pre-
sented for a multilayered mushroom-type metamaterial.
Section III focuses on the analysis of negative refraction in
the proposed configurations with the plane-wave incidence
and Gaussian beam excitation. In particular, the structure
with air gaps is of practical interest, wherein the angle of
negative refraction can be controlled. The conclusions are
summarized in Sec. IV.
II. HOMOGENIZATION OF MULTILAYERED
MUSHROOM-TYPE METAMATERIAL
The multilayered mushroom structure is formed by the
grids of metallic square patches separated by dielectric slabs
perforated with metallic pins (vias) connected to the metallic
elements. The geometry of the structure with a transverse
magnetic (TM) plane-wave incidence is shown in Fig. 1.
Here, a is the period of the patches and the vias, g is the gap
between the patches, h is the thickness of the dielectric layer
between the patch arrays, ehis the permittivity of the dielec-
tric slab, and r0is the radius of the vias.
In our analytical model, the dielectric slabs perforated
with vias are modeled as WM slabs, and the patch arrays are
treated as homogenized surfaces with the capacitive grid im-
pedance obtained from the effective circuit parameters for sub-
wavelength elements.17For completeness, we consider two
different homogenization models (nonlocal and local) for the
wire medium as described in the sections to follow, with the
aim of demonstrating that in the proposed multilayered config-
uration (Fig. 1) the SD effects are significantly reduced. A
time dependence of the form ejxtis assumed and suppressed.
A. Nonlocal homogenization model
For long wavelengths the WM can be characterized by a
spatially-dispersive model of a uniaxial material with the
effective relative permittivity along the vias (Refs. 15 and
16, and references therein):
enonloc
zz
ðx;kzÞ ¼ eh 1 ?
k2
p
k2
h? k2
z
!
;
(1)
where kh¼ k0
k0¼ x=c is the free space wave number, x is the angular
frequency, c is the speed of light in vacuum, and kzis the
z-component of the wave vector~k ¼ kx;0;kz
kp is the plasma wave number defined in Ref. 11 as kp
¼
that the plasma wave number depends on the period and on
the radius of the vias. The dependence of the permittivity
on the wave vector is a consequence of the fact that
the macroscopic electric displacement cannot be linked to
the macroscopic electric field through a local relation. The
nonlocal model predicts the propagation of TM and trans-
verse electromagnetic (TEM) fields in the wire medium.18
Suppose that a plane wave with the y-polarized magnetic
field (TM polarization) is incident at an angle hi(with the
plane of incidence chosen as the x-z plane) on the structure
as shown in Fig. 1. Following Ref. 18, the electric and mag-
netic fields in the WM slab [with the nonlocal dielectric
function (Eq. 1)] can be expressed in terms of waves propa-
gating along opposite directions with respect to the z-axis:
ffiffiffiffi
eh
p
is the wave number in the host material,
ðÞ. In Eq. 1,
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð2p=a2Þ=½logða=2pr0Þ þ 0:5275?
p
. It should be noted
g0Hy¼ Aþ
TMeþcTMzþ A?
þ B?
j
ehk0½cTMðAþ
þ cTEMðBþ
kx
eTM
ffiffiffiffi
pþ k2
TM-polarization, and g0is the impedance of free space. Simi-
larly, the fields associated with the reflected and transmitted
waves in the air regions (above and below the multilayered
structure) are obtained in terms of the reflection and transmis-
sion coefficients, R and T. At the patch grid interfaces (at the
planes z ¼ z0¼ 0;?h;?2h;::::;?L) the tangential electric
and magnetic fields can be related via sheet admittance,
TMe?cTMzþ Bþ
TEMeþcTEMz
TEMe?cTEMz;
(2)
Ex¼
TMeþcTMz? A?
TEMeþcTEMz? B?
?
, cTM¼
ofthe
xÞ is the relative permittivity along the vias for
TMe?cTMzÞ
TEMe?cTEMzÞ?;
(3)
Ez¼ ?
zzk0
p
Aþ
TMeþcTMzþ A?
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
wave
TMe?cTMz
?;
(4)
where cTEM¼ jk0
is the
¼ ehk2
eh
k2
pþ k2
x? k2
0eh
vector
q
, kx¼ k0sinhi
~k,
x-component
x=ðk2
eTM
zz
Ex¼ ?1
yg
ðHyjz¼zþ
0? Hyjz¼z?
0Þ
(5)
with the Ex-component of the electric field continuous across
the patch grid,
Exjz¼zþ
0¼ Exjz¼z?
0:
(6)
In Eq. 5, ygis the normalized effective grid admittance
of the patch array,17
yg¼?j1
g0
eqsk02a
pln csc
pg
2a
??hi
(7)
FIG. 1. (Color online) 3D view of a multilayered mushroom-type metama-
terial formed by periodically attaching metallic patches to an array of paral-
lel wires.
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Page 3

with eqs¼ ehþ 1
grids and eqs¼ ehfor all the internal grids. In order to find
the unknown amplitudes A6
TM and TEM fields in Eqs. 2–4 besides the boundary condi-
tions Eqs. 5 and 6 at the patch interfaces, an additional
boundary condition (ABC) is required at the connection of
WM to the metallic patches. Following Refs. 13, 14 and 19,
the ABC is associated with the zero charge density at the
connection of metallic pins to the metallic elements of the
capacitive patch arrays [equivalently for the microscopic
current at the connection point, dIðzÞ=dz ¼ 0], and is
expressed in terms of the macroscopic field components,
ðÞ=2 for the lower and the upper external
TMand B6
TEMassociated with the
k0ehdEz
dzþ kxg0
dHy
dz
¼ 0:
(8)
Using Eqs. 5, 6, and the ABC (Eq. 8), the reflected (at
the upper interface) and the transmitted (at the lower inter-
face) fields of the entire multilayered mushroom structure
(Fig. 1) are related in the matrix form:
?
Ex
g0Hy
?
lower interface
¼ MG?
Ex
g0Hy
??
upper interface
(9)
where MG is the global transfer matrix, which takes into
account the product of the transfer matrices across the plane
of metallic patches and for the propagation across the region
in between two adjacent patch arrays (as WM slab). The ana-
lytical expressions of the transfer matrices are given in the
Appendix.
The reflection and transmission coefficients, R and T, of
the multilayered mushroom structure can be easily obtained
from Eq. 9 by solving the following matrix equation
"
1
MG?
jc0
k0
?1
#
R þ
jc0
k0
"#
T ¼ MG?
jc0
k0
1
"#
:
(10)
B. Local homogenization model
In the local homogenization model, the WM slab (as a
uniaxial continuous ENG material) is characterized for long
wavelengths by the classical Drude dispersion model, which
does not take into account SD effects (Refs. 13 and 14, and
references therein):
eloc
zzðxÞ ¼ eh 1 ?k2
p
k2
h
!
:
(11)
This approximation is valid when the current along the
vias is uniform (or for long vias at low frequencies when the
WM can be characterized as a material with extreme anisot-
ropy).18This assumption is justified because both ends of the
vias are connected to the metallic elements of the patch
arrays, and the charge is distributed over the surface of the
metallic patches. Therefore, the charge density is approxi-
mately zero at the connection points and along the vias, and
the field is nearly uniform in the WM slab.18Within the local
model formalism, the amplitudes of the electric and
magnetic field components in the WM slab are expressed as
follows:
g0Hy¼ Hþeþczþ H?e?cz;
jc
ehk0
(12)
Ex¼
Hþeþcz? H?e?cz
ðÞ;
(13)
where c is the propagation constant in the WM slab along
the direction of the vias given in Refs. 13 and 14
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
eloc
zz
c ¼
k2
x
? k2
h
s
:
(14)
The local model takes into account only the effect of fre-
quency dispersion in the WM slab, and treats the WM as an
ENG uniaxial continuous material below the plasma fre-
quency. The local model does not require an ABC, as it does
not take into account the SD effects in the WM.
The transmitted and reflected fields are related in a simi-
lar matrix form Eqs. 9 and 10, satisfying the classical bound-
ary conditions for tangential electric and magnetic field
components at interfaces (Eqs. 5 and 6). The analytical
expressions of the transfer matrices used in the local model
are given in the Appendix. It should be noted that the local
homogenization model may predict accurately the response
of the structure when the SD effects are significantly reduced
(it will be shown in Section III, that this is the case in a mul-
tilayered mushroom structure).
III. RESULTS AND DISCUSSION
In this section, the transmission properties of the mush-
room-type metamaterials are studied under the plane-wave
incidence, using both the nonlocal and local homogenization
models. The negative refraction effect is characterized from the
obtained transmission properties. We consider two multilay-
ered mushroom-type metamaterials: the first configuration is as
shown in Fig. 1, and the second one is formed by the inclusion
of air gaps (without vias) in between two-layered (paired)
mushrooms (with the geometry shown in Fig. 6 later in the pa-
per). The motivation for considering the latter configuration is
that it may be much easier to fabricate, and provides further
degrees of freedom in the design of the metamaterial. In addi-
tion, the phenomenon of negative refraction is confirmed with
full-wave commercial software that models the incidence of a
Gaussian beam on a finite width metamaterial slab.
A. Multilayered mushroom-type metamaterial
As a first example, we consider a multilayered mush-
room structure formed by five identical patch arrays sepa-
rated by four dielectric layers perforated with vias (the
geometry of a generic structure is shown in Fig. 1). Each
patch array has the period a¼2 mm and gap g¼0.2 mm,
and each dielectric slab is of thickness 2 mm with permittiv-
ity 10.2. The period of the vias is 2 mm with a radius of 0.05
mm. The plasma frequency (fp=
approximately at 12.15 GHz. The transmission properties
(magnitude and phase) of the structure based on the local and
ffiffiffiffi
eh
p
) of the WM slab is
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nonlocal homogenization models for a TM-polarized plane
wave incident at 45 degrees are shown in Fig. 2. It is seen that
the results of the two models are in good agreement with the
full-wave simulations results obtained with CST Microwave
StudioTM,21especially in the region below the plasma fre-
quency. In the vicinity of the plasma frequency the local
model shows spurious resonances in a very narrow frequency
band. The spurious resonances appear because of the singular-
ity in Eq. 14, where eloc
mushroom structures studied in Refs. 13 and 14, it has been
shown that the spatial dispersion effects in the wire medium
can be suppressed (or significantly reduced) by loading vias
with a capacitive grid of metallic patches. This results in a
nearly uniform current along the vias, i.e., d=dz ? 0 or in the
spectral domain kz? 0. Under this condition the nonlocal
dielectric function Eq. 1 reduces to the local dielectric func-
tion Eq. 11. Consistent with these findings and with Refs. 15
and 16, the results of Fig. 2 (showing an excellent agreement
of the results of nonlocal and local homogenization models,
even above the plasma frequency) support that for the consid-
ered geometry of a multilayered metamaterial (Fig. 1) the
effects of spatial dispersion are suppressed and below the
plasma frequency it behaves as a uniaxial continuous ENG
material loaded with patch arrays.
In order to study the emergence of negative refraction in
the multilayered mushroom structure we use the formalism
proposed in Ref. 7, which is based on the analysis of the var-
zz¼ 0 at the plasma frequency. For the
iation in the phase of T x;kx
plane wave characterized by the transverse wave number kx)
of the metamaterial slab with the incident angle hi. Specifi-
cally, it was shown in Ref. 7, that for an arbitrary material
slab excited by a quasiplane wave, apart from the transmis-
sion magnitude, the field at the output plane differs from the
field at the input plane by a spatial shift D [see inset in Fig.
3(b)], given by D ¼ du=dkx, where u ¼ argT. The transmis-
sion angle can be obtained as ht¼ tan?1D=L (L is the thick-
ness of the planar material slab). Thus, negative refraction
occurs when D is negative, i.e., when u decreases with the
angle of incidence hi. It should be noted that the calculation
of spatial shift is more accurate when there is a smooth varia-
tion in the magnitude of the transfer function T x;kx
Figure 3 demonstrates the behavior of the magnitude
and phase of the transmission coefficient versus the angle of
incidence at the frequency of 11 GHz (eloc
be seen that the phase of transmission coefficient decreases
with an increase in the incident angle, which indicates
unequivocally the emergence of negative refraction. The
total transmission occurs at the incident angle of 32.73
degrees, and the calculated spatial shift (using finite differen-
ces for the calculation of du=dkx) at this angle is
D ¼ ?1:02k0 (k0 is the free-space wavelength at 11 GHz)
with the electrical thickness of the metamaterial slab equal
to L ¼ 0:29k0. The calculated transmission angle is ?73.8
degrees, thus demonstrating a strong negative refraction.
This shows that the multilayered mushroom-type structure
ðÞ (transmission coefficient for a
ðÞ.
zz¼ ?2:23). It can
FIG. 3. (Color online) Comparison of local (blue dashed lines), nonlocal
(green dot-dashed lines), and full-wave CST results (orange full lines) for
the five-layered (five patch arrays with four WM slabs) structure as a func-
tion of the incident angle of a TM-polarized plane wave. (a) Magnitude of
the transmission coefficient. (b) Phase of the transmission coefficient.
FIG. 2. (Color online) Comparison of local (blue dashed lines), nonlocal
(green dot-dashed lines), and full-wave CST results (orange full lines) for
the five-layered (five patch arrays with four WM slabs) structure excited by
a TM-polarized plane wave incident at 45 degrees. (a) Magnitude of the
transmission coefficient. (b) Phase of the transmission coefficient.
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enables negative refraction at an interface with air, when the
effects of spatial dispersion in the WM are suppressed.
Next, we consider the dependence of the negative refrac-
tion on the number of layers of the multilayered structure.
Specifically, we have calculated the spatial shift D and the
transmission angle htas a function of the incident angle hi,
for a different number of identical layers of the mushroom
structure (with the same dimensions as considered in the pre-
vious example). Figure 4 shows the analytical results based
on the local homogenization model at the frequency of 11
GHz. It is evident that there is an increase in the absolute
value of the spatial shift with the increase in the number of
layers (substantial increase in the overall length of the meta-
material). However, there is no significant change in the angle
of transmission (negative refraction). The calculated D and ht
for a different number of patch arrays with the incident angle
tuned to achieve maximum transmission are listed in Table I.
It is worth considering the effect of the negative refrac-
tion with respect to the operating frequency. We have calcu-
lated the spatial shift D and the transmission angle htas a
function of the incidence angle hiat different frequencies for
the six-layered structure (six identical patch arrays with five
identical WM slabs) with the same dimensions used in the
previous examples. The results of the local homogenization
model are depicted in Fig. 5. It can be seen that the phenom-
enon of negative refraction is observed over a wide fre-
quency band below the plasma frequency.
Although negative refraction is observed over a wide fre-
quency band, its strength becomes gradually weaker with the
TABLE I. Characterization of the negative refraction with an increase in
the number of identical layers.
N grids
D=k0
L=k0
hi(deg)
ht(deg)
2
3
4
5
6
7
?0.22
?0.45
?0.7
?1.0
?1.30
?1.73
0.074
0.148
0.22
0.29
0.364
0.438
22.96
29.09
31.33
32.73
34.68
35.76
?71.4
?71.9
?72.55
?73.8
?74.62
?75.79
FIG. 4. (Color online) (a) Spatial shift D and (b) transmission angle htas a
function of the incident angle hiof a TM-polarized plane wave calculated
for the multilayered structure with a different number of layers.
FIG. 5. (Color online) (a) Spatial shift D and (b) transmission angle htfor
the six-layered (six patch arrays and five WM slabs) structure as a function
of the incident angle hiof a TM-polarized plane wave calculated at different
frequencies.
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Page 6

decrease in the frequency of operation. For instance, at the
frequency of 11 GHz, the maximum negative refraction angle
is ?74.62 degrees, and it decreases to ?34.52 degrees with
the decrease in the operating frequency to 8 GHz. However,
when one operates above the plasma frequency (eloc
mushroom structure exhibits positive refraction. It is apparent
that the frequency range where one can observe the negative
refraction can be shifted by changing the plasma frequency.
It should be noted that the negative refraction properties of
the proposed multilayered mushroom structure can be con-
trolled by the geometrical parameters. For example (results
are not reported here), an increase in the radius of the vias
increases the plasma resonance frequency, and, therefore, in
order to operate in the negative refraction regime (close to
the plasma resonance) the frequency has to be increased. This
may, however, result in the conditions when the homogeniza-
tion is no longer valid. Also, the requirement that the patches
in different layers are connected through the metallic vias
creates obvious difficulties in the practical realization of a
structure with a large number of layers (due to technological
difficulties in the alignment of the layers in the stacked struc-
ture, and in the realization of long vias of small radius).
In the next section we propose an alternative structure that
overcomes these problems and provides one extra degree of
freedom to control the negative refraction angle of the meta-
material without changing its structural properties.
zz> 0), the
B. Multilayered mushroom-type metamaterial with air
gaps
Here we consider a mushroom-type metamaterial with air
gaps, as shown in Fig. 6. The structure is formed by several
two-sided mushroom slabs (with two symmetric patch arrays
connected with vias) separated by air gaps. Here, a is the pe-
riod of the patches and the vias, g is the gap between the
patches, h is the thickness of the dielectric layer between the
patch arrays, ehis the permittivity of the dielectric slab, hais
the thickness of the air gap, and r0is the radius of the vias.
We consider the case of a structure formed by two
mushroom slabs with an air gap. The dimensions are the
same as used in the previous examples, and the thickness of
the air gap is 2 mm. The transmission response of the struc-
ture based on the local and nonlocal homogenization models
for the TM-polarized plane wave incident at 45 degrees is
shown in Fig. 7. It is seen that there is a good agreement
between the results of the two models. Also, the homogeni-
zation results agree reasonably well with the full-wave simu-
lation results obtained with HFSS,22especially in the region
below the plasma frequency.
We have characterized the negative refraction using the
same procedure as in the previous section. It can be seen from
Fig. 8, that at the operating frequency of 11 GHz, there is a
monotonic decrease in the angle u ¼ argT with the variation
in the incidence angle, except for large incident angles corre-
sponding to the rapid change in the transmission magnitude.
This clearly indicates that the multilayered mushroom meta-
material with air gaps enables negative refraction. The calcu-
lated negative spatial shift at the incident angle of 23.3
degrees corresponding to the transmission maximum is
FIG. 6. (Color online) 3D view of the mushroom-type metamaterial formed
by including the air gap (without vias) in between two-layered (paired)
mushrooms.
FIG. 7. (Color online) Comparison of local (blue dashed lines), nonlocal
(green dot-dashed lines), and full-wave HFSS results (orange full lines) for
the multilayered mushroom structure with an air gap excited by a TM-polar-
ized plane wave incident at 45 degrees. (a) Magnitude of the transmission
coefficient. (b) Phase of the transmission coefficient.
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Page 7

D ¼ ?0:35k0. The electrical length of the multilayered struc-
ture at 11 GHz is L ¼ 0:22k0, and the calculated transmission
angle is ?58.15 degrees. It is interesting, that despite the pres-
ence of an air region (characterized by a positive refraction),
the structure still exhibits quite significant negative refraction.
In order to further characterize the dependence of nega-
tive refraction on the thickness of the air gap ha, we have cal-
culated the spatial shift D and the transmission angle htas a
function of the incidence angle hi, at the operating frequency
of 11 GHz. The results are depicted in Fig. 9, and are
obtained based on the local homogenization model. It can be
observed that there is a steady decrease in the calculated neg-
ative spatial shift with the increase in the thickness of the air
gap. This is due to the increased positive spatial shift in the
air region (with the increase of the air gap). Moreover, there
is a significant decrease in the negative refraction angle. The
calculated spatial shift D and the angle of transmission htfor
different values of the air gap hawith the incident angle hiof
the transmission maximum are given in Table II. It is evident
that the incident angle remains almost the same, however,
the corresponding transmission angle decreases significantly.
Consequently, it is possible to control the negative refraction
angle by varying the thickness of the air gap. The proposed
geometry is of great practical interest, because of the ease in
fabrication. Interestingly, the structure exhibits negative
refraction only for moderate angles (<45 degrees) of inci-
dence. Considering the fact that the positive spatial shift in
the air region is dependent on the angle of incidence, for
large incident angles the suffered positive spatial shift in the
FIG. 8. (Color online) Comparison of local (blue dashed lines), nonlocal
(green dot-dashed lines), and full-wave CST results (orange full lines) for
two double-sided mushroom slabs separated by an air gap as a function of
the incident angle of a TM-polarized plane wave. (a) Magnitude of the trans-
mission coefficient. (b) Phase of the transmission coefficient.
FIG. 9. (Color online) (a) Spatial shift D and (b) transmission angle htas a
function of the incident angle hiof a TM-polarized plane wave calculated
for the multilayered structure with the varying thickness of the air gap ha.
TABLE II. Characterization of the negative refraction as a function of the
thickness of the air gap ha.
ha(mm)
D=k0
L=k0
hi(deg)
ht(deg)
2
4
6
8
10
?0.35
?0.32
?0.29
?0.26
?0.22
0.22
0.29
0.37
0.44
0.51
23.3
22.9
22.8
22.1
21.9
?58.15
?47.73
?38.59
?30.61
?23.32
044901-7 Kaipa, Yakovlev, and Silveirinha J. Appl. Phys. 109, 044901 (2011)
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Page 8

air region dominates the negative spatial shift in the wire me-
dium, thus creating a positive refraction.
It is interesting to see if the negative refraction is
observed over a wide frequency band. We have calculated D
and htas a function of the incident angle at different frequen-
cies, and the analytical results based on the local model are
shown in Fig. 10. It can be seen that there is a rapid decrease
in the strength of the negative refraction with the decrease in
the operating frequency (below the plasma frequency). In
fact (as discussed in the case without air gaps), the negative
refraction becomes gradually weaker away from the plasma
frequency, i.e., the absolute value of the negative spatial shift
decreases. Consequently, in this case the positive spatial shift
in the air region dominates, thus reducing the frequency
band for the emergence of negative refraction.
C. Gaussian beam excitation
To further confirm the predicted phenomenon of negative
refraction based on the homogenization models, we have
simulated the response of the metamaterial structures excited
by a Gaussian beam using CST Microwave Studio.21In the
simulation setup, the structure is assumed to be periodic along
y with the period a (equal to the period of the patch array),
and is finite along x with the width Wx¼ 90a. The considered
Gaussian beam field distribution is independent on the y-coor-
dinate. We excite simultaneously 10 adjacent waveguide
ports, with the electric width of the each port being 0:3k0at
the design frequency. The amplitude and phase of each wave-
guide are chosen such that the wave radiated by the port array
mimics the profile of a Gaussian beam and propagates along a
desired direction hiin the x-z plane. The cases of mushroom
slabs with and without air gaps are considered, with the same
geometrical parameters used in the previous examples. In the
simulation, the effects of losses are taken into account: the
metallic componentsare
(r ¼ 5:8 ? 107S=m), and a loss tangent of tand ¼ 0:0015 is
considered in dielectric substrates (RT/duroid 6010LM).
The results obtained with the CST Microwave Studio
are shown in Fig. 11. The snapshot (t ¼ 0) of the amplitude
of the magnetic field Hyof the Gaussian beam incident at 19
degrees is shown in Fig. 11(a). The Gaussian beam-waist is
approximately 1:6k0at the operating frequency of 11 GHz.
Figure 11(b) shows the snapshot of the amplitude of Hyin
the vicinity of the metamaterial structure with a 2 mm air
gap (formed by two mushroom slabs with an air gap as
shown in Fig. 6) illuminated by the Gaussian beam incident
at 19 degrees. It can be observed that the transmitted beam
suffers a negative spatial shift, demonstrating a significant
negative refraction inside of the metamaterial. Similar results
are depicted in Fig. 11(d) with the Gaussian beam incident at
30 degrees. The simulation results are qualitatively consist-
ent with the theoretical values (predicted by the homogeniza-
tion models) for the spatial shift D ¼ ?0:31k0
D ¼ ?0:33k0, calculated with the incident angles of 19
degrees and 30 degrees, respectively. Figure 11(c) depicts
the case of the Gaussian beam incident at 19 degrees on the
metamaterial structure formed by three mushroom slabs with
two air gaps. The theoretical value of the spatial shift for the
configuration in Fig. 11(c) is D ¼ ?0:44k0, while that for the
case shown in Fig. 11(b) is D ¼ ?0:31k0. However, it should
be noted that the negative refraction angle remains almost the
same [with the predicted theoretical values of ?54.63 degrees
and ?50.63 degrees for the cases (b) and (c), respectively].
The magnetic field in the vicinity of the five-layered
structure without air gaps (with the geometry shown in Fig. 1)
for the Gaussian beam incident at 32 degrees is depicted in
Fig. 11(e). It is seen that the transmitted beam suffers a large
negative spatial shift, thus exhibiting a strong negative refrac-
tion, and the simulation results are consistent with the ones
reported in Refs. 15 and 16. The negative spatial shift in the
metamaterial is a significant fraction of the wavelength.
It should be noted that due to computational limitations the
Gaussian beam cannot be treated exactly as a quasiplane wave
(since its beam-width is marginally larger than 1:5k0). Thus, the
analytical results based on the homogenization models are qual-
itatively accurate but quantitatively approximate in modeling a
realistic finite metamaterial with the Gaussian beam excitation.
modeledascopper metal
and
IV. CONCLUSION
In this paper, we investigated multilayered mushroom-
type structures as bulk metamaterials which enable strong
FIG. 10. (Color online) (a) Spatial shift and (b) transmission angle for the
multilayered structure with an air gap of 2 mm as a function of incident
angle of a TM-polarized plane wave calculated at different frequencies.
044901-8 Kaipa, Yakovlev, and SilveirinhaJ. Appl. Phys. 109, 044901 (2011)
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Page 9

negative refraction. The transmission properties of the meta-
materials are studied based on the local and nonlocal homog-
enization models.
Consistent with Refs. 15 and 16, it was shown that the
multilayered mushroom-type metamaterial behaves as a local
(with no spatial dispersion) uniaxial ENG material periodi-
cally loaded with patch arrays. The negative refraction is
observed over a wide frequency band below the plasma fre-
quency, and is accurately predicted by the homogenization
models. The strength of the negative refraction decreases
gradually when we operate away from the plasma frequency.
In addition, we proposed a modified structure where the
mushroom slabs are separated by air gaps. It was shown that
this configuration also exhibits significant negative refraction,
and enables the control of the negative transmission angle by
varying the thickness of the air gap without changing the
structural properties of the metamaterial. Such configuration
is of great practical importance because of the ease in fabrica-
tion. It is planned to obtain the experimental verification
of the proposed configurations, which is the scope of future
work.
The observed phenomenon of negative refraction was
qualitatively verified with the Gaussian beam excitation
using CST Microwave Studio.
APPENDIX
The tangential electric and magnetic fields given by
Eqs. 2 and 3 across the patch interfaces (at the planes
z ¼ z0¼ 0;?h;?2h;:::::;?L) can be related in the matrix
form using the two-sided impedance boundary conditions
(Eqs. 5 and 6),
FIG. 11. (Color online) CST Microwave Studio simulation results showing the snapshot (t¼0) of the magnetic field Hyexcited by a Gaussian beam: (a) inci-
dent beam with hi¼19 degrees (no metamaterial slab), (b) two mushroom slabs with an air gap for an angle of incidence hi¼19 degrees, (c) three mushroom
slabs with two air gaps for an angle of incidence hi¼19 degrees, (d) two mushroom slabs with an air gap for an angle of incidence hi¼30 degrees, and (e)
five-layered structure (without air gaps with the geometry shown in Fig. 1) for an angle of incidence hi¼32 degrees. The operating frequency for all the cases
is 11 GHz and the thickness of the air gap is 2 mm.
044901-9Kaipa, Yakovlev, and SilveirinhaJ. Appl. Phys. 109, 044901 (2011)
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Page 10

Ex
g0Hy
??
z¼z?
0
¼ Q
Ex
g0Hy
??
z¼zþ
0
;
(A1)
where Q is the transfer matrix across the plane of patches,
?
In Eq. A2, ygis the normalized admittance of the patch grids
given by Eq. 7. The transfer matrix for the propagation in the
WM slab between the two adjacent patch grids used in the
nonlocal homogenization model is obtained by substituting
Eqs. 2 and 4 in the ABC (Eq. 8), and relating the tangential
Q ¼
1
yg
0
1
?
:
(A2)
electric and magnetic field components at the plane
z ¼ z0? h
Ref. 20, the transfer matrix is as follows:
?
where the matrix P is
?
with the matrix elements:
ðÞþto the fields at the plane z ¼ z?
0. Following
Ex
g0Hy
?
z¼ z0?h
ðÞþ¼ P ?
Ex
g0Hy
??
z¼zþ
0
;
(A3)
P ¼
p11
p21
p12
p22
?
(A4)
p11¼ p22¼
eh? eTM
zz
??cTMsinh cTMh
eh? eTM
ð
zz
Þcosh cTEMh
?cTMsinh cTMh
jcTEMcTMsinh cTMh
eh? eTM
ð Þ þ eTM
Þ þ eTM
zzcTEMcosh cTMh
zzcTEMsinh cTEMh
ðÞsinh cTEMh
Þ
ðÞ
?
ðð
;
(A5)
p12¼ ?1
k0
ðÞsinh cTEMh
Þ þ eTM
ðÞ
zz
??cTMsinh cTMh
ð
zzcTEMsinh cTEMh
ðÞ;
(A6)
p21¼ jk0
2 eh? eTM
eh? eTM
zz
?cTMsinh cTMh
??eTM
zz?1 þ cosh cTMh
ð
ðÞcosh cTEMh
zzcTEMsinh cTEMh
ðÞ½?
zz
?
Þ þ eTM
ðÞþ
sinh cTEMh
ð
eh? eTM
Þsinh cTMh
?cTMsinh cTMh
ðÞ
ð
eh? eTM
Þ þ eTM
zz
??2cTM
zzcTEMsinh cTEMh
cTEM
þ eTM
zz
??2cTEM
cTM
Þ
??
zz
?
ð
8
>
>
>
>
:
<
9
>
>
>
>
;
=
:
(A7)
The global transfer matrix for the entire multilayered
structure can be obtained as a product of the corresponding
transfer matrices,
MG¼ Q0? P ? Q ? ? ? ?P ? Q0;
(A8)
where Q0is the transfer matrix across the plane of patches
for the upper and the lower external grids and Q is the trans-
fer matrix for all internal grids.
The global transfer matrix for the local model is the
same as that of the nonlocal model, except for the transfer
matrix P. The transfer matrix P (for the propagation in the
WM slab), is obtained by matching the fields at the interfaces
z ¼ z0? h
expressed as follows:
ðÞþand z ¼ z?
0in a similar form as Eq. A3, and is
P ¼
cosh ch
?
ðÞ
c
jk0eh
??
sinh ch
ðÞ
jk0eh
c
?
sinh ch
ðÞ
cosh ch
ðÞ
2
664
3
775:
(A9)
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044901-10Kaipa, Yakovlev, and Silveirinha J. Appl. Phys. 109, 044901 (2011)
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp