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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)


14:54:24

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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)


14:54:24

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).



14:54:24

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).



14:54:24

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�.



14:54:24

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.



14:54:24

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.



14:54:24

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.



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.



14:54:24

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.



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)

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Mushroom-type structures with the wires connected through ...yakovlev/Pdfs/1.4954676.pdf · with mushroom-type structures (high impedance surface and bi-layer) with diodes inserted