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Mushroom-type structures with the wires connected through diodes: Theory andapplicationsAli Forouzmand, Chandra S. R. Kaipa, and Alexander B. Yakovlev Citation: Journal of Applied Physics 120, 015303 (2016); doi: 10.1063/1.4954676 View online: http://dx.doi.org/10.1063/1.4954676 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/120/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Temperature dependent simulation of diamond depleted Schottky PIN diodes J. Appl. Phys. 119, 225703 (2016); 10.1063/1.4953385 Characterization and modeling of n - n Si ∕ Si C heterojunction diodes J. Appl. Phys. 102, 014505 (2007); 10.1063/1.2752148 Current impulse response of thin InP p + ‐ i ‐ n + diodes using full band structure Monte Carlo method J. Appl. Phys. 101, 044502 (2007); 10.1063/1.2434827 Avalanche noise measurement in thin Si p + -i-n + diodes Appl. Phys. Lett. 76, 3926 (2000); 10.1063/1.126823 On the extraction of linear and nonlinear physical parameters in nonideal diodes J. Appl. Phys. 85, 6873 (1999); 10.1063/1.370206
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Mushroom-type structures with the wires connected through diodes: Theoryand applications
Ali Forouzmand,a) Chandra S. R. Kaipa,b) and Alexander B. Yakovlevc)
Department of Electrical Engineering, University of Mississippi, University, Mississippi 38677-1848, USA
(Received 1 March 2016; accepted 8 June 2016; published online 6 July 2016)
In this paper, we establish a general formalism to quantify the interaction of electromagnetic waves
with mushroom-type structures (high impedance surface and bi-layer) with diodes inserted along
the direction of the wires. The analysis is carried out using the nonlocal homogenization model for
the mushroom structure with the generalized additional boundary conditions at the connection of
the wires to diodes. We calculate numerically the magnitude and phase of the reflected/transmitted
fields in the presence of an ideal and realistic PIN diodes. It is observed that the reflection/transmis-
sion characteristics of the mushroom-type structures can be controlled by tuning the working states
of the integrated PIN diodes. We realize a structure with a multi-diode switch to minimize the
undesired transmission for a particular incident angle. In addition, a dual-band subwavelength
imaging lens is designed based on the resonant amplification of evanescent waves, wherein the
operating frequency can be tuned by changing the states of the PIN diodes. The analytical results
are verified with the full-wave electromagnetic solver CST Microwave Studio, showing a good
agreement. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4954676]
I. INTRODUCTION
In recent years, electronically tunable metasurfaces
became attractive for applications such as beam shaping and
steering, three dimensional holography, absorbers, and
reflectarrays, among others. By controlling the properties of
the surface impedance of a metasurface, the reflection phase
characteristics, surface-wave propagation, and leaky wave
radiation can be tailored based on the requirements in desired
applications. As an example, in conventional fixed beam
reflectarray antennas, phasing of the scattered field in order
to realize the expected radiation pattern is obtained by vary-
ing the physical characteristics of each element, for example,
metallic patches with a variable size1 or elements having a
variable rotation angle.2 In Ref. 3, electronic control of the
element has been shown by loading the radiating edge of a
patch antenna with a varactor diode. By varying the reversed
applied voltage, the varactor’s capacitance is changed which
results in the control of the reflection phase. In addition, a
reflectarray utilizing an electronically tunable impedance
surface has been realized by using non-resonant subwave-
length elements connected by varactor diodes.4 These elec-
tronically tunable metasurfaces find extensive applications in
the design of reconfigurable antennas with electronical beam
steering and high directivity.5
Microwave interference is an important issue in various
applications due to the surface currents induced on metallic
surfaces. Conventional methods to suppress surface currents
are lossy coating,6,7 reactive surface such as a high-
impedance surface,8,9 and absorbing materials.10–13 These
traditional techniques intrinsically have several disadvan-
tages such as being heavy, bulky, reducing the performance,
and operating in a narrow bandwidth. In order to surmount
these restrictions, the concept of circuit based nonlinear
metasurface absorbers has been studied in Refs. 14–16. The
nonlinear absorbing behavior is obtained by employing
diodes integrated into the metasurface. The diodes rectify
high-power signals to produce a static field, whose energy is
stored in capacitors and then dissipated with resistors. It
allows the high-power absorption properties of a surface to
be decoupled from its low-power scattering behavior.
Here, we consider mushroom-type structures (high
impedance surface and bi-layer) with diodes inserted at the
center along the direction of the wires, with a typical geome-
try shown in Fig. 1. A nonlocal homogenization model is uti-
lized, such that the wire medium (WM) is modeled as a
uniaxial anisotropic material characterized by a nonlocal
dielectric function with the generalized additional boundary
conditions (GABCs) at the connection of the wires to the
diodes.17 Following the effective-medium approach,18,19 the
reflection/transmission properties can be obtained by applying
the classical and additional boundary conditions. It is observed
that the frequency response of the mushroom structures can be
modified by switching the operating states of the PIN diodes.
As a result, we realize a structure with a multi-diode switch to
minimize the undesired transmission for particular incident
angle. It should be noted that the structures proposed in the
present work are different from those considered in the afore-
mentioned works, wherein the diodes are placed on the top of
the surface connecting the metallic plates (patches/strips).14–16
It is observed that apart from electronically controlling the
reflection and transmission characteristics, the present design
allows for a dual-band subwavelength imaging lens in which
the operating frequency can be tuned by changing the states of
the PIN diodes. Although, we restrict our study to PIN diodes
in this paper, the presented methodology can be utilized to
incorporate varactor diodes. Moreover, it is possible to
a)[email protected])[email protected])[email protected]
0021-8979/2016/120(1)/015303/11/$30.00 Published by AIP Publishing.120, 015303-1
JOURNAL OF APPLIED PHYSICS 120, 015303 (2016)
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consider diodes with series connected lumped loads in analyti-
cal approach which provides more degrees of freedom to
achieve a desired tunable response.
The paper is organized as follows. In Sec. II, we present
the reflection characteristics of a grounded mushroom-type
structure with wires connected through diodes based on the
nonlocal homogenization model. In Sec. III, the transmission/
reflection response of a bi-layer structure is studied with the
PIN diodes in ON and OFF states. A structure with a multi-
diode switch is designed to reduce the undesired transmission
for a specific incident angle. In Sec. IV, the dispersion behav-
ior of even modes of a bi-layer mushroom-type structure
loaded with diodes is calculated in ON and OFF states. The
performance of proposed structure is analyzed in the presence
of a magnetic line source and a dual-band sub-diffraction
imaging lens which operates at two distinct frequencies in
ON and OFF states is designed. The conclusions are drawn in
Sec. V. A time dependence of the form ejxt is assumed andsuppressed.
II. NONLOCAL MODEL FOR MUSHROOM-TYPESTRUCRURE WITH DIODES
Here, we consider a grounded mushroom structure with
the wires connected through diodes (MWDs) as shown in
Fig. 1(b). The patch array is at z ¼ 0, D is the spacingbetween the wires (lattice constant), r0 is the radius of thewires, eh is the dielectric permittivity of the slab, and h is thethickness of the structure. The wires are connected to the
patches and the ground plane at z ¼ 0 and z ¼ �h, respec-tively, and through the diodes at z ¼ �h=2. It should benoted that the position of diodes can be changed arbitrarily
along the direction of the wires. The reflection properties of
the structure with a transverse magnetic (TM) plane-wave
incidence can be obtained by applying the classical and addi-
tional boundary conditions20–25 which are described in detail
in Appendix A. In addition, the complexity of problem due
to the presence of diodes dictates the necessity of employing
GABCs at the connection of the wires through the diodes
which will be carefully addressed and discussed in this sec-
tion. A rigorous analytical solution for this problem can be
obtained if and only if a generalized additional boundary
condition is taken into account for the microscopic current at
the connection of the wires to diodes. The proper GABCs for
the ideal and realistic diodes are described as follows.
A. Ideal diode
An ideal diode acts like a perfect insulator when it is
reverse biased (positive voltage is supplied to the cathode
and negative voltage to the anode), i.e., it does not allow the
flow of current. Therefore, we assume an open circuit (O.C.)
at the connection of the diode to the wires in OFF state. For
this ideal case, the following GABC’s should be satisfied:
I2ðzÞj z¼�h=2þ ¼ I3ðzÞj z¼�h=2� ¼ 0: (1)
It should be noted that the position of diodes can be arbi-
trarily changed along the wires. When the diode is forward
biased (positive voltage is supplied to the anode and negative
voltage to the cathode), diode acts like a perfect conductor
and consequently it conducts current. In ON state, the diode
behaves as a short circuit (S.C.), and the appropriate GABCs
can be written as
I2ðzÞj z¼�h=2þ ¼ I3ðzÞj z¼�h=2� ; (2)
dI2 zð Þdz
���� z¼�h=2þ ¼ dI3 zð Þdz���� z¼�h=2� : (3)
In order to examine the validation of (1)–(3) and
compatibility of the presented formulation in Appendix A
for arbitrary located diodes along the wires, we consider a
mushroom structure with the wires connected to the ground
through the diodes located at z ¼ �h=3. The dimensionsof the structure are as follows: D¼ 2 mm, g¼ 0.2 mm,r0¼ 0.05 mm, eh¼ 10.2, h¼ 1.5 mm, and hi¼ 60. Fig. 2demonstrates the reflection phase characteristics for ideal
diodes in OFF (Fig. 2(a)) and ON (Fig. 2(b)) states as a func-
tion of frequency. In the full-wave simulation, it is assumed
that the wire is connected to diode through a gap of 0.1 mm
(as shown in Fig. 2(c)). It is observed that the reflection char-
acteristics for an ideal diode are in good agreement with
CST Microwave Studio simulation.26 As a starting point, the
investigation of diodes in the simplest case can provide intui-
tive understanding about general characteristics of unit-cell,
but it is evident that the realistic diodes do not have com-
pletely the same characteristics.
B. Realistic diode
In practice, diodes cannot conduct infinite current due to
the presence of internal resistance and diodes cannot be per-
fect insulators when reverse biased (positive voltage to the
cathode), so they will conduct some leakage current. When
diode is in ON state, it offers a small resistance (typically a
few ohms due to ohmic losses) and it exhibits a small
FIG. 1. Geometry of (a) bi-layer and (b) grounded mushroom structures
with the wires connected through diodes excited by an obliquely incident
TM-polarized plane wave.
015303-2 Forouzmand, Kaipa, and Yakovlev J. Appl. Phys. 120, 015303 (2016)
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capacitance in the OFF state. These facts should be incorpo-
rated in the GABCs for the efficient modeling of the struc-
ture. In order to satisfy the GABCs in the presence of
realistic diode, the currents (I2 and I3) should be continuous[similar to Eq. (2)] and the derivative of currents should be
related via the impedance of diode as follows:
dI3 zð Þdz
���z¼�h=2�
� dI2 zð Þdz
���z¼�h=2þ
¼ jxCwireZdiodeI2 zð Þj z¼�h=2þ ; (4)
where Cwire¼ 2pehe0= log ða2=4r0ða� r0ÞÞ is the capacitanceper unit length of the wire medium and Zdiode is the imped-ance of the diode. Following the circuit model, the diode can
be realized by a very low resistance and capacitance in the
ON and OFF states, respectively. Here, we restrict our stud-
ies on two PIN diodes, namely, MADP-000907-14020 and
MA4GP907. The circuit model and the values of the lumped
elements can be extracted from the data sheets for the corre-
sponding diodes in Refs. 27 and 28. Diode is not a linear
device and the performance depends on the applied voltage,
current, and operating frequency. Our studies show that these
PIN diodes, in a wide range of frequencies (f ¼ 2–15 GHz),have resistance and capacitance which are almost constant.
In this paper, we consider the resistance and capacitance of
diode as R ¼ 3 X and C ¼ 0:025 pF, respectively.In order to simulate the realistic diode in the CST
Microwave Studio software, we utilized the physical pack-
age dimensions of diode which are mentioned in Refs. 27
and 28. Figure 3 illustrates the geometry of a unit cell of the
mushroom structure with the wire connected through the
realistic diode. The realistic diode as shown in Fig. 3 consists
of two parallel copper sheets which are connected by a
lumped load (capacitor in OFF state and resistor in ON
state).
Figure 4 represents the reflection phase characteristics
for a mushroom structure with the realistic diodes in ON
state versus frequency in the range of 0–18 GHz. To obtain
the analytical and full-wave simulation results for reflection
from a grounded mushroom structure with the same struc-
tural parameters as in the previous case, the diode is modeled
similar to Fig. 3 and it is inserted in the wire through a gap
of 0:73 mm. In the ON and OFF states, the lumped load ischosen equal to R ¼ 3 X and C ¼ 0:025 pF, respectively.The difference between analytical (dashed-dotted line) and
CST (dashed line) results as shown in Fig. 4 is due to the
non-uniformity of current and charge distributions along the
wire in the presence of realistic diode. In order to carefully
address this physical phenomenon, the gap which is filled by
diode should be characterized by parasitic loads in analytical
FIG. 3. Geometry of a unit cell of the mushroom structure with the wire con-
nected through the realistic diode.
FIG. 2. The analytical and full-wave
simulation results for the reflection
phase characteristics of a grounded
MWDs in (a) ON and (b) OFF states
with the ideal diodes at z ¼ �h=3excited by an obliquely incident TM-
polarized plane wave at 60�. (c) Thegeometry of a unit-cell in the presence
of an ideal diode (the gap size is
0.1 mm).
015303-3 Forouzmand, Kaipa, and Yakovlev J. Appl. Phys. 120, 015303 (2016)
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approach. Therefore, the correction terms such as the para-
sitic capacitance Cpar and parasitic inductance Lpar should betaken into account through an updated effective diode im-
pedance, Zdiode;eff , as
Zdiode;eff ¼ jxLpar þ1
jxCpar þ1
Zdiode
: (5)
The value of effective diode impedance is estimated by
curve fitting of analytical and CST simulation results. The
appropriate parasitic loads to characterize the gap are parasitic
capacitance of Cpar ¼ 0:02 pF and parasitic inductance ofLpar ¼ 0:1 nH. It can be seen that the homogenization resultswith the consideration of parasitic loads (solid line in Fig. 4)
are in good agreement with the full-wave numerical results.
It is worth noting that to ensure that all the elements are
electrically net connected, the unit cells can be linked by the
connecting wires (wire grid) inserted in between. The wire
grid made of high resistive material can be used as a biasing
wire, which would be transparent to the impinging electromag-
netic waves. Then, the structure can be tuned by a DC voltage
bias.29,30 It should be noted that for practical realization one
requires to use a diode pair in anti-parallel such that there is a
current flow along the wires (in both directions) for forward
and backward traveling waves. Also, it is possible to switch
the working states of the diodes depending on the power of the
incident waves. At low power, the diodes are in the OFF state
and act as an open circuit. However, with an increase in power,
the diodes are forward biased, and the vias of the metasurface
transform from the open circuit to the short circuit.
As a supplementary study, the transmission/reflection
response of a bi-layer mushroom structure loaded with ideal
and realistic diodes is discussed in Appendix B. It should be
noted that the methodology presented in this paper can be
used to incorporate varactor diodes and diodes with series
connected lumped loads to have a desired tunable response.
Here, two applications of the mushroom structure with
diodes will be presented along with the numerical results.
III. CONTROL OVER TRANSMISSION ANDREFLECTION
The transmission response of a bi-layer mushroom struc-
ture with realistic diodes is studied (with the geometry
shown in Fig. 1(a)). The diodes are located at z ¼ �h=2 and
the physical parameters of the structure are as follows: D ¼2 mm, g ¼ 0:2 mm, r0 ¼ 0:05 mm, eh ¼ 10:2, h ¼ 2 mm,and hi ¼ 60�. The analytical approach details to calculate thereflection and transmission coefficients are provided in
Appendix B [Eqs. (B6) and (B7)]. Figures 5 and 6 demon-
strate the transmission amplitude and phase of a bi-layer
mushroom structure with the realistic diodes in OFF (Fig. 5)
and ON (Fig. 6) states as a function of frequency. It should
be mentioned that the full-wave simulation results are
achieved under careful consideration of realistic parameters
of diodes and the same parasitic capacitance
(Cpar ¼ 0:02 pF) and inductance (Lpar ¼ 0:1 nH) have beentaken into account in the analytical approach. This reliable
agreement between analytical and full-wave simulation
results validates this claim that the parasitic loads only
depend on the physical parameters of realistic diode and
once they are determined can be utilized for other designs of
a mushroom structure.
The transmission behavior of the unit-cell is changed by
switching the diodes from OFF to ON state. The structure
has a high transmission amplitude (�1) at frequenciesaround 10:64 GHz when the diodes are in OFF state. On theother hand, in ON state, the transmission characteristic is
completely changed in vicinity of 10:64 GHz and it has alow transmission amplitude (�0:2). Therefore, it is possibleto design a structure with a multi-diode switch to minimize
the undesired transmission for a particular incident angle.
This property can be utilized in order to obtain optimum iso-
lation for the region below the structure. It should be men-
tioned that the system is narrow-band in the attainment of
desired performance. In Fig. 7, a row of 20 unit-cells is simu-
lated under an obliquely incident TM-polarized plane wave
(on the structure from the top) with the incidence angle of
60� at the operating frequency of f ¼ 10:64 GHz. The struc-ture is assumed periodic along the y�direction, and theabsorbing boundary condition is applied to the x� andz�direction. Figs. 7(a) and 7(b) represent the field distribu-tion of a bi-layer mushroom structure in ON and OFF states,
respectively. For practical realization, the device can be fab-
ricated with sufficient number of unit cells (same geometry
as in the simulation model) in x and y directions withthe exception of a diode pair in anti-parallel as discussed
before and the measurements can be carried out as discussed
in Ref. 16.
FIG. 4. The analytical and full-wave
simulation results for the reflection
phase characteristics of a grounded
mushroom-type structures with realis-
tic diodes at z ¼ �h=3 excited by anobliquely incident TM-polarized plane
wave at 60�. The dashed-dotted line isthe analytical results without consider-
ation of parasitic loads. The solid line
is obtained with Cpar ¼ 0:02 pF andLpar ¼ 0:1 nH which is validated byCST simulation results (dashed line).
015303-4 Forouzmand, Kaipa, and Yakovlev J. Appl. Phys. 120, 015303 (2016)
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The main mechanism behind this switchable device is
that the state of diodes will change the structure topology,
and consequently its transmission properties. The metasur-
face can transform from one state to another only by control-
ling the power level of the incident beam or the DC voltage
of the biasing circuit. Here, in this case for the biasing cir-
cuit, we require two wire grids, made of high resistive mate-
rial, acting as a biasing wire above and below the diodes. At
low power, the diodes appear as an open circuit and the
structure behaves as simple parallel capacitive sheets and
when the signal level surpasses the turn-on voltage of diodes,
they connect the patches to each other behaving as a uniaxial
epsilon negative (ENG) material.
IV. SUBWAVELENGTH IMAGING
An infinite magnetic line source is oriented along the y�direction and placed at a distance d from the upper interfaceof the structure. The geometry is depicted in Fig. 8. The mag-
netic current density of the line source is Jm ¼ I0dðz� dÞdðxÞŷ, resulting in the incident magnetic field given by
FIG. 5. The comparison of analytical
and full-wave simulation results for
transmission amplitude and phase of a
bi-layer mushroom structure with real-
istic diodes in OFF state excited by an
obliquely incident TM-polarized plane
wave at 60�.
FIG. 6. The comparison of analytical
and full-wave simulation results for
transmission amplitude and phase of
the bi-layer mushroom structure with
realistic diodes in ON state excited by
an obliquely incident TM-polarized
plane wave at 60�.
015303-5 Forouzmand, Kaipa, and Yakovlev J. Appl. Phys. 120, 015303 (2016)
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H x; zð Þ ¼I0k
20
jxl0
1
4jH 2ð Þ0 k0qð Þ
� �ŷ; (6)
where q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ ðz� dÞ2
qand H
ð2Þ0 ðk0qÞ is the zero-order
Hankel function of the second kind. The transmitted mag-
netic field at a distance d from the lower interface of thestructure can be obtained as a Sommerfeld-type integral
Hy xð Þ ¼I0k
20
jpxl0
ð10
1
2c0e�c0 2dð ÞT x; kxð Þcos kxxð Þdkx; (7)
where c0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2x � k20
pis the propagation constant in free
space and Tðx; kxÞ is the transfer function of the structure[defined in Appendix B, see Eq. (B7)].
First, we are interested in the design of a sub-diffraction
lens with the capability of recovering the source details at
the image plane in ON state and preventing the subwave-
length imaging in OFF state. In other words, the impinging
evanescent waves interact resonantly with the surface plas-
mons on the interfaces and experience an amplification along
the lens when the diodes are in ON state. Conversely, the
near fields decay rapidly and all subwavelength information
will be lost at a short distance away from the source in OFF
state. Second, a dual-band subwavelength imaging lens
which can operate at two distinct frequencies only by switch-
ing the diodes from ON to OFF state is of particular interest.
To determine the proper operating frequency regime for
subwavelength imaging, the dispersion behavior of the TMx
surface waves of the MWDs in both ON and OFF states is
studied. As it has been discussed in Ref. 31, the significant
resonances of the structure are associated with perfect elec-
tric conductor (PEC) symmetry (even modes). Therefore, we
restrict our analysis to the dispersion relation obtained by
applying the PEC plane at z ¼ �h=2 (shown in Fig. 15). Thedispersion of the surface waves is calculated by finding the
FIG. 7. Bi-layer mushroom structure
excited by an obliquely incident TM-
polarized plane wave at 60� in (a) OFFand (b) ON states. Transmission of
structure decreases drastically when
the state of diodes is changed from
OFF to ON at f ¼ 10:64 GHz.
FIG. 8. Geometry of MWDs excited by a magnetic line source placed at a
distance d from the upper interface, with the image plane at a distance dfrom the lower interface: (a) 3D view and (b) cross-section.
015303-6 Forouzmand, Kaipa, and Yakovlev J. Appl. Phys. 120, 015303 (2016)
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complex roots of the dispersion function [denominator of the
reflection coefficient in Eq. (B4)]. Figure 9 represents the
dispersion behavior of the normalized propagation constant
Reðkx=k0Þ of the TMx even guided modes of MWDs in OFF/ON state with/without consideration of the parasitic loads.
The structural parameters are as follows: D¼ 2 mm,g¼ 0.2 mm, r0¼ 0.05 mm, er ¼ 10.2, and h¼ 2 mm. In theON state, at low frequencies (f < 8 GHz), the real part ofðkx=k0Þ is close to 1 indicating that the proper bound forwardmode (the phase and group velocities have the same direc-
tion) interacts weakly with the WM slab and propagates pri-
marily in the air region in the vicinity of patches. In this
regime, the real part of the proper bound backward mode has
large value. This mode is highly dispersive and strongly
interacts with the structure. (The field is primarily concen-
trated below the patches in the wire medium.) At the fre-
quencies around 9:05 GHz, the forward and backward modesjoin together (turning point) and a stopband occurs for the
first TMx surface bound mode.
Figure 9 shows that by switching the diodes from ON to
OFF state, the dispersion behavior will be changed drasti-
cally and the MWDs behave similar to two parallel patches
in which the evanescent waves could not be effectively
enhanced. Due to the fact that the performance of the lens is
highly sensitive to the frequency dispersion, the parasitic
loads (realistic diodes) are considered in our analysis and the
effect of parasitic loads on the dispersion behavior has been
plotted in Fig. 9.
Figure 10 shows the square normalized amplitude of the
magnetic field profiles calculated at the image plane as a
function of x=k at the operating frequency of f ¼ 9:05 GHzwhen the diodes are in ON and OFF states. It is assumed that
the magnetic line source is located at d ¼ 0:05k. In Fig. 10,the blue solid line is obtained by the numerical integration of
the Sommerfeld integral in Eq. (7). According to the half-
power beam width (HPBW) criterion, the resolution is
approximately 0:22k. The black solid line represents themagnetic field profile for the propagation in free space with
the resolution of 0:58k. The resolution of the proposed struc-ture is approximately 2.65 times better than the resolution in
free space. The red dashed line and the green dotted line rep-
resent the performance of the proposed lens studied by using
the electromagnetic simulator CST Microwave Studio in ON
and OFF states, respectively. The resolution of k=4:54 forthe structure has been obtained analytically and verified with
the full-wave simulation.
In CST Microwave Studio, the magnetic line source is
modeled by a current-carrying square loop and the structure
is assumed periodic along the y�direction and the width ofthe slab has been fixed at 1:8k0 along the x�direction. Themetallic wires are modeled as the copper metals (r ¼ 5:8�107 S
m) and the effect of ohmic losses is taken into account.
A snapshot of the magnetic field (Hy) in the x� z plane cal-culated using CST is shown in Fig. 11 at f ¼ 9:05 GHz. InFig. 11(a), the image can be observed at the lower interface
of the structure. The resolution of the image is k=4:54 andit is nearly insensitive to the effect of losses. In contrast,
the lens is unable to restore the field distribution of the
object plane when the diodes are in OFF states as shown
in Fig. 11(b).
Next, we investigate a structure with the same structural
parameters as in the previous example except that the thick-
ness of lens is increased from 2 mm to 10 mm, and dielectric
material is replaced by air (eh ¼ 1). In addition, we utilize alumped inductive load which is connected in series with
diode in order to increase the resolution of the lens. The
square normalized amplitude of the magnetic field profile
FIG. 9. Dispersion behavior of the real part of the normalized propagation
constant Reðkx=k0Þ of the even TMx modes of MWDs in OFF/ON statewith/without consideration of the parasitic loads.
FIG. 10. The square normalized ampli-
tude of the magnetic field Hy calcu-lated at the image plane for f ¼ 9:05GHz when the diodes are in ON and
OFF states.
015303-7 Forouzmand, Kaipa, and Yakovlev J. Appl. Phys. 120, 015303 (2016)
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-
calculated at the image plane versus x=k at the operating fre-quency of 6.699 GHz is shown in Fig. 12. The magnetic line
source is placed at d ¼ 0:05k. The blue solid line is obtainedby the analytical technique at the image plane. The resolu-
tion based on the HPBW criterion is 0:124k. The dashed lineshows the full-wave simulation result achieved by using
CST Microwave Studio, and the HPBW resolution is equal
to 0:124k. The black solid line corresponds to the magneticfield propagation in free space with the resolution of 0:38k.The resolution of the proposed structure improves nearly 3
times in comparison to free space. According to HPBW cri-
terion, the resolution of k=8 has been obtained analyticallyand validated by the full-wave simulation. As an important
point, it should be mentioned that the better performance of
the lens in terms of the resolution in comparison with the
case of h¼ 2 mm is because the dispersion curve of the struc-ture has the required value of Re(kx=k0) at the resonance fre-quency which leads to the higher and more uniform
transmission response as discussed in Refs. 32 and 33. The
dotted line shows the behavior of the lens when the diode is
OFF at the same operating frequency of 6.699 GHz. Figures
13(a) and 13(b) represent the magnetic field distributions in
ON and OFF states, respectively. In Fig. 13(b), all subwave-
length information is lost and the image is not formed at the
image plane. On the other hand, Figure 13(a) illustrates that
the MWDs in ON state can effectively transmit the near field
information of the source to the image plane.
In contrast to the other well-known subwavelength
imaging devices which operate at single frequency, this lens
can operate at two distinct frequencies. In order to clarify the
possibility of achieving subwavelength imaging in both ON
and OFF states, Fig. 14(c) shows the dispersion behavior of
the normalized propagation constant ðkx=k0Þ of the evenTMx modes of MWDs in ON and OFF states. By changing
the diodes state, the dispersion relation shifts to the higher
frequencies and the operating frequency increases from
6:699 GHz to 9:96 GHz. In Figs. 14(a) and 14(b), the blueand orange solid lines have been obtained by the numerical
integration of the Sommerfeld integral in Eq. (7) and the
HPBWs are equal to 0:124k and 0:25k. Therefore, the reso-lution better than k=4 for both dual operating frequencies hasbeen obtained.
FIG. 11. CST simulation results for the
magnetic field distribution Hy ofMWDs in (a) ON state, the lens
restores the field distribution from the
object plane and (b) OFF state, show-
ing that the subwavelength information
is not transmitted to the image plane.
The magnetic line source is located at
a distance of d ¼ 0:05k from the upperinterface of the structure, and the
image plane is located at the same dis-
tance from the lower interface.
FIG. 12. The square normalized ampli-
tude of the magnetic field Hy calcu-lated at the image plane for f ¼ 6:699GHz when the diodes are in ON and
OFF states. The black curve represents
the field profile when the structure is
absent. The blue curve is the field pro-
file in the presence of the structure.
The dashed and dotted lines corre-
spond to the CST Microwave Studio
results.
015303-8 Forouzmand, Kaipa, and Yakovlev J. Appl. Phys. 120, 015303 (2016)
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14:54:24
-
V. CONCLUSION
The reflection and transmission characteristics of mush-
room configurations have been studied using the effective-
medium approach with the generalized ABCs at the insertion
of diodes in the wires. It is shown that the frequency
response of the mushroom structures can be changed by
controlling the operating states of the PIN diodes. The pro-
posed configuration based on diodes enables to minimize the
undesired transmission for particular incident angle. In addi-
tion, a dual-band subwavelength imaging lens is designed
based on the enhancement of evanescent waves where the
operating frequency of lens can be controlled by switching
the state of PIN diodes. The analytical results are verified
with the full-wave electromagnetic solver CST Microwave
Studio, showing good agreement.
ACKNOWLEDGMENTS
This work has been partially supported by the NASA
EPSCoR Award No. NNX13AB31A. The authors are also
grateful to the reviewers for their valuable comments.
APPENDIX A: NONLOCAL HOMOGENIZATION MODELFOR A GROUNDED MUSHROOM STRUCTURE WITHTHE WIRES CONNECTED THROUGH DIODES
In the nonlocal homogenization model, a wire medium
is characterized by a uniaxial anisotropic material with the
effective permittivity of ezz ¼ eh½1� k2p=ðk2h � k2z Þ�, wherekh ¼ k0
ffiffiffiffiehp
is the wave number of the host medium, k0 ¼x=c is the free space wave number, x is the angular fre-quency, c is the speed of light in vacuum, kz is the z-compo-
nent of the wave vector ~k ¼ ðkx; 0; kzÞ, and kp is the plasmawave number as kp ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2p=a2Þ=ðlnða=2pr0Þ þ 0:5275Þ
p.
It is assumed that TM plane wave propagating in x� zplane is incident on MWDs at an angle hi. A TM-polarizedplane wave excites both transverse electromagnetic (TEM)
and TM modes in the homogenized wire medium (WM)
slab. The tangential electric and magnetic fields in the air
region above the structure (region I) and inside the WM slab
�h=2 < z < 0 (region II) and �h < z < �h=2 (region III),as shown in Fig. 1(b), can be written as follows:
FIG. 13. CST simulation results for the
magnetic field distribution Hy ofMWDs in (a) ON state and (b) OFF
state.
FIG. 14. The square normalized amplitude of the magnetic field Hy calcu-lated at the image plane for (a) f ¼ 6:699 GHz and (b) f ¼ 9:96 GHz. (c)Dispersion behavior of the real part of the normalized propagation constant
Reðkx=k0Þ of the even TMx modes of a MWDs in ON and OFF states. Thedashed and dashed-dotted lines correspond to the proper bound forward
mode in OFF and ON states, respectively.
015303-9 Forouzmand, Kaipa, and Yakovlev J. Appl. Phys. 120, 015303 (2016)
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Region I:
H 1ð Þy ¼ ec0z � Re�c0z; E 1ð Þx ¼
�c0jxe0
ec0z þ Re�c0zð Þ; (A1)
where R is the electric-field reflection coefficient from agrounded mushroom structure, c0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2x � k20
p, and kx ¼ k0
sin hi is the x� component of the wave vector ~k ¼ ðkx; 0; kzÞ.Region II:
H 2ð Þy ¼ AþTM2ecTM zþh2ð Þ þ A�TM2e�cTM zþ
h2ð Þ þ BþTEM2ecTEM zþ
h2ð Þ
þ BþTEM2e�cTEM zþh2ð Þ;
E 2ð Þx ¼�1
jxe0ehcTM A
þTM2e
cTM zþh2ð Þ � A�TM2e�cTM zþh2ð Þ
� ��
þcTEM BþTEM2ecTEM zþh2ð Þ � BþTEM2e�cTEM zþ
h2ð Þ
� ��;
E 2ð Þz ¼�kxg0eTMzz k0
AþTM2ecTM zþh2ð Þ þ A�TM2e�cTM zþ
h2ð Þ
� �: (A2)
Region III:
H 3ð Þy ¼ AþTM3ecTM zþhð Þ þ A�TM3e�cTM zþh
ð Þ
þBþTEM3ecTEM zþhð Þ þ BþTEM3e�cTEM zþh
ð Þ;
E 3ð Þx ¼�1
jxe0ehcTM A
þTM3e
cTM zþhð Þ � A�TM3e�cTM zþhð Þ
� ��
þcTEM BþTEM3ecTEM zþhð Þ � BþTEM3e�cTEM zþh
ð Þ� ��
;
E 3ð Þz ¼�kxg0eTMzz k0
AþTM3ecTM zþhð Þ þ A�TM3e�cTM zþh
ð Þ� �
: (A3)
A6TM3, B6TEM3; A
6TM2, and B
6TEM2 are the amplitudes of the
extraordinary TM and transmission-line TEM modes in the
WM slab, cTEM ¼ jkh ¼ jk0ffiffiffiffiehp
, eTMzz ¼ ehk2x=ðk2p þ k2xÞ is therelative effective permittivity for TM polarization, and
cTM ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2p þ k2x � k2h
q. In (A1)–(A3), R, A6TM3, B
6TEM3; A
6TM2,
and B6TEM2 are the unknown amplitude coefficients which canbe obtained by enforcing the following boundary conditions.
The two-sided impedance boundary condition should be
applied at the air-patches interface (z ¼ 0) such that the tan-gential electric and magnetic fields can be related via the
grid admittance as follows:
E 1ð Þx j z¼0þ ¼ E 2ð Þx j z¼0� ¼
1
YgH 2ð Þy j z¼0� � H 1
ð Þy j z¼0þ
h i; (A4)
where Yg is the grid admittance of the patch array
Yg ¼ �j1
g0
eh þ 12
k02D
pln csc
pg2D
�� �: (A5)
The tangential components of the electric fields must van-
ish at the ground plane ðz ¼ �hÞ. It has been shown that anadditional boundary condition (ABC) is required at the connec-
tion of the wires to the metallic patches and perfect electric con-
ductor.20–24 Following Ref. 25, the microscopic ABC for the
wire current, I(z) at the connection of the wires to the patches
(z ¼ 0�) and the ground plane (z ¼ �hþ) can be written as
dI2 zð Þdz
���z¼0�¼ dI3 zð Þ
dz
���z¼�hþ
¼ 0; (A6)
where I2ðzÞ and I3ðzÞ correspond to the microscopic wirecurrents in the medium above and below the diode, respec-
tively, at the plane z ¼ �h=2. The tangential components ofthe electric and magnetic fields are continuous at the inter-
face z ¼ �h=2.
APPENDIX B: NONLOCAL HOMOGENIZATION MODELFOR A BI-LAYER MUSHROOM STRUCTURE LOADEDWITH DIODES
To determine the transmission/reflection response of the
bi-layer mushroom structure loaded with diodes which is
illuminated by a TM-polarized plane wave, the even/odd
excitation technique is utilized. By considering the perfect
electric conductor (PEC) and the perfect magnetic conductor
(PMC) at the center of the wires (z ¼ �h=2), the even andodd responses of the structure can be obtained. Fig. 15 dem-
onstrates the cross-section view in the presence of PEC/PMC
symmetries.
The total magnetic fields in the air region above the
structure (region 1) and in the WM slab (region 2), as shown
in Fig. 15, can be expressed as follows:
Hð1Þy ¼ ðec0z þ Reven=odde�c0zÞe�jkxx;Hð2Þy ¼ ðAþTMecTM z þ A�TMe�cTM z
þBþTEMecTEM z þ B�TEMe�cTEM zÞe�jkxx; (B1)
where A6TM and B6TEM are the amplitudes of the extraordinary
TM and transmission-line TEM modes in the WM slab, and
Reven=odd is the reflection coefficient of the even and oddexcitations. The unknown coefficients (Reven=odd, A
6TM, and
B6TEM) can be determined by enforcing the appropriateboundary conditions which are discussed in the manuscript.
In addition, the discontinuities in the microscopic wire
current distribution at the connection of wires to the patches
and to the ground plane through diodes should be taken into
account through following GABCs:
dI zð Þdz� jxCwireZdiodeð ÞI zð Þ
#z¼�h=2þ
¼ 0;
24 (B2)
FIG. 15. Cross-section view of the mushroom structure with diodes excited
by an obliquely incident TM-polarized plane wave with the consideration of
PEC/PMC at the symmetry plane.
015303-10 Forouzmand, Kaipa, and Yakovlev J. Appl. Phys. 120, 015303 (2016)
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14:54:24
-
dI zð Þdzþ Cwire
Cpatch
�I zð Þ#
z¼0�¼ 0;
24 (B3)
where Cpatch ¼ pe0ðeh þ 1Þða� gÞ= log½sec p g2a
�
� is the ca-pacitance of the patch in a periodic array of patches. The mi-
croscopic wire current can be expressed in terms of the bulk
electromagnetic fields as IðzÞ ¼ �ja2½ðk0eh=g0ÞEz þ kxHy�.By enforcing the aforementioned boundary conditions,
the closed-form expression of the reflection coefficient for
PEC symmetry (even excitation) can be obtained by31
K ¼ cTMsinh cTMhð Þcos khhð Þ � kh sin khhð Þ
� eheTMzz� 1
�cosh cTMhð Þ þ
ehcTMsinh cTMhð ÞeTMzz jxCwireZdiode;eff
" #;
M ¼ 2 eh � eTMzz
�
þ cosh cTMhð Þjehkh
xCwireZdiode;effsin khhð Þ
�
þ eheh
eTMzz� 2
�þ 2eTMzz
�cos khhð Þ
#
þ eh � eTMzz
�
sinh cTMhð ÞcTEMcTMþ cTM
cTEM
�j sin khhð Þ
�
þ ehcTMeTMzz jxCwireZdiode;eff
cos khhð Þ�;
Reven ¼jk0 � g0c0Ygð ÞK � jk0c0Mjk0 þ g0c0Ygð ÞK þ jk0c0M
: (B4)
The closed-form expression of the reflection coefficient
for the PMC symmetry (odd excitation) can be derived using
a similar approach31
Rodd ¼ �
ehk2x tanh cTMhð ÞcTM k2x þ k2p
� þ ehk2p tan khhð Þ
kh k2x þ k2p
� � 1
c0þ j Ygg0
k0
�ehk2x tanh cTMhð ÞcTM k2x þ k2p
� þ ehk2p tan khhð Þ
kh k2x þ k2p
� þ 1
c0� j Ygg0
k0
� :
(B5)
The transmission/reflection response of the bi-layer
structure can be obtained by the superposition principle as
follows:
R ¼ 12
Reven þ Roddð Þ; (B6)
T ¼ 12
Reven � Roddð Þ: (B7)
1D. M. Pozar, S. D. Targonski, and H. D. Syrigos, IEEE Trans. Antennas
Propag. 45, 287 (1997).
2J. Huang and R. J. Pogorzelski, IEEE Trans. Antennas Propag. 46, 650(1998).
3L. Boccia, G. Amendola, and G. D. Massa, in Proceedings of the IEEEInternational Symposium on Antennas and Propagation, June (2004), Vol.4, p. 3927.
4D. F. Sievenpiper, J. H. Schaffner, H. J. Song, R. Y. Loo, and G.
Tangonan, IEEE Trans. Antennas Propag. 51, 2713 (2003).5D. F. Sievenpiper and J. H. Schaffner, Electron. Lett. 38, 1237 (2002).6E. Knott, Radar Cross Section Measurements (Springer Science &Business Media, 2012).
7B. A. Munk, Frequency Selective Surfaces: Theory and Design (Wiley,New York, 2000).
8N. Engheta and R. Ziolkowski, Metamaterials Physics and EngineeringExplorations (IEEE Press, Piscataway, NJ, 2006).
9D. Sievenpiper, Z. Lijun, R. F. J. Broas, N. G. Alexopolous, and E.
Yablonovitch, IEEE Trans. Microwave Theory Tech. 47, 2059(1999).
10N. I. Landy, S. Sajiuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla,
Phys. Rev. Lett. 100, 207402 (2008).11D. J. Kern and D. H. Werner, Microwave Opt. Technol. Lett. 38, 61
(2003).12H. Wakatsuchi, S. Greedy, C. Christopoulos, and J. Paul, Opt. Express 18,
22187 (2010).13H. Wakatsuchi and C. Christopoulos, Appl. Phys. Lett. 98, 221105
(2011).14D. F. Sievenpiper, Antennas Wireless Propag. Lett. 10, 1516 (2011).15H. Wakatsuchi, S. Kim, J. J. Rushton, and D. F. Sievenpiper, Appl. Phys.
Lett. 102, 214103 (2013).16S. Kim, H. Wakatsuchi, J. J. Rushton, and D. F. Sievenpiper, Appl. Phys.
Lett. 108, 041903 (2016).17S. I. Maslovski, T. A. Morgado, M. G. Silveirinha, C. S. R. Kaipa, and A.
B. Yakovlev, New J. Phys. 12, 113047 (2010).18O. Luukkonen, M. G. Silveirinha, A. B. Yakovlev, C. R. Simovski, I. S.
Nefedov, and S. A. Tretyakov, IEEE Trans. Microwave Theory Tech. 57,2692 (2009).
19C. S. R. Kaipa, A. B. Yakovlev, S. I. Malovski, and M. G. Silveirinha,
IEEE Antennas Wireless Propag. Lett. 10, 1503 (2011).20M. G. Silveirinha, IEEE Trans. Antennas Propag. 54, 1766 (2006).21M. G. Silveirinha, C. A. Fernandes, and J. R. Costa, New J. Phys. 10,
053011 (2008).22A. B. Yakovlev, Y. R. Padooru, G. W. Hanson, A. Mafi, and S. Karbasi,
IEEE Trans. Microwave Theory Tech. 59, 527 (2011).23A. B. Yakovlev, M. G. Silveirinha, O. Luukkonen, C. R. Simovski, I. S.
Nefedov, and S. A. Tretyakov, IEEE Trans. Microwave Theory Tech. 57,2700 (2009).
24M. G. Silveirinha, C. A. Fernandes, and J. R. Costa, IEEE Trans. Antennas
Propag. 56, 405 (2008).25M. G. Silveirinha and C. A. Fernandes, IEEE Trans. Microwave Theory
Tech. 53, 1418 (2005).26See http://www.cst.com for CST Microwave Studio 2014, CST GmbH.27See http://www.macomtech.com/datasheets/MADP-000907-14020.pdf for
Data sheet of MADP-000907-14020.28See http://www.macomtech.com/datasheets/MA4GP907.pdf for Data
sheet of MA4GP907 GaAs flip chip PIN.29M. Wang, C. Hu, M. Pu, C. Huang, X. Ma, and X. Luo, Electron. Lett. 48,
1002 (2012).30X. Ma, W. Pan, C. Huang, M. Pu, Y. Wang, B. Zhao, J. Cui, C. Wang, and
X. Luo, Adv. Opt. Mater. 2, 945 (2014).31C. S. Kaipa, A. B. Yakovlev, S. I. Maslovski, and M. G. Silveirinha, Phys.
Rev. B 86, 155103 (2012).32A. Forouzmand and A. B. Yakovlev, AIP Adv. 5, 077108 (2015).33A. Forouzmand, H. M. Bernety, and A. B. Yakovlev, Phys. Rev. B 92,
085402 (2015).
015303-11 Forouzmand, Kaipa, and Yakovlev J. Appl. Phys. 120, 015303 (2016)
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14:54:24
http://dx.doi.org/10.1109/8.560348http://dx.doi.org/10.1109/8.560348http://dx.doi.org/10.1109/8.668907http://dx.doi.org/10.1109/TAP.2003.817558http://dx.doi.org/10.1049/el:20020863http://dx.doi.org/10.1109/22.798001http://dx.doi.org/10.1103/PhysRevLett.100.207402http://dx.doi.org/10.1002/mop.10971http://dx.doi.org/10.1364/OE.18.022187http://dx.doi.org/10.1063/1.3597628http://dx.doi.org/10.1109/LAWP.2011.2182593http://dx.doi.org/10.1063/1.4809535http://dx.doi.org/10.1063/1.4809535http://dx.doi.org/10.1063/1.4940712http://dx.doi.org/10.1063/1.4940712http://dx.doi.org/10.1088/1367-2630/12/11/113047http://dx.doi.org/10.1109/TMTT.2009.2032458http://dx.doi.org/10.1109/LAWP.2011.2180694http://dx.doi.org/10.1109/TAP.2006.875920http://dx.doi.org/10.1088/1367-2630/10/5/053011http://dx.doi.org/10.1109/TMTT.2010.2090358http://dx.doi.org/10.1109/TMTT.2009.2031933http://dx.doi.org/10.1109/TAP.2007.915442http://dx.doi.org/10.1109/TAP.2007.915442http://dx.doi.org/10.1109/TMTT.2005.845128http://dx.doi.org/10.1109/TMTT.2005.845128http://www.cst.comhttp://www.macomtech.com/datasheets/MADP-000907-14020.pdfhttp://www.macomtech.com/datasheets/MA4GP907.pdfhttp://dx.doi.org/10.1049/el.2012.1318http://dx.doi.org/10.1002/adom.201400212http://dx.doi.org/10.1103/PhysRevB.86.155103http://dx.doi.org/10.1103/PhysRevB.86.155103http://dx.doi.org/10.1063/1.4926399http://dx.doi.org/10.1103/PhysRevB.92.085402