Partial focusing by a bulk metamaterial formed by a periodically loadedwire medium with impedance insertionsChandra S. R. Kaipa and Alexander B. Yakovlev Citation: J. Appl. Phys. 112, 124902 (2012); doi: 10.1063/1.4769875 View online: http://dx.doi.org/10.1063/1.4769875 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v112/i12 Published by the American Institute of Physics. Related ArticlesCoupled electromagnetic TE-TM wave propagation in a layer with Kerr nonlinearity J. Math. Phys. 53, 123530 (2012) Experimental investigations of the TE11 mode radiation from a relativistic magnetron with diffraction output Phys. Plasmas 19, 113108 (2012) Omnidirectional photonic band gap enlarged by one-dimensional ternary unmagnetized plasma photonic crystalsbased on a new Fibonacci quasiperiodic structure Phys. Plasmas 19, 112102 (2012) An oversized X-band transit radiation oscillator Appl. Phys. Lett. 101, 173504 (2012) Path derivation for a wave scattered model to estimate height correlation function of rough surfaces Appl. Phys. Lett. 101, 141601 (2012) Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

Partial focusing by a bulk metamaterial formed by a periodically loaded wiremedium with impedance insertions

Chandra S. R. Kaipaa) and Alexander B. Yakovlevb)

Department of Electrical Engineering, The University of Mississippi, University, Mississippi 38677-1848, USA

(Received 20 September 2012; accepted 19 November 2012; published online 19 December 2012)

In this paper, a uniaxial wire medium periodically loaded with metallic patches and lumped

impedance insertions is proposed for partial focusing of electromagnetic radiation due to a

magnetic line source. The analysis is based on the nonlocal homogenization model for a bi-layer

mushroom structure with generalized additional boundary conditions for loaded vias, and it is

extended to a multilayered configuration with the transfer matrix approach. The proposed

structure exhibits a high transmission and is nearly insensitive to the losses. The analytical results

are validated against full-wave numerical simulations. VC 2012 American Institute of Physics.

[http://dx.doi.org/10.1063/1.4769875]

I. INTRODUCTION

Artificial media formed by an array of long parallel

conducting thin wires (so-called wire media) has been used

at microwaves for a long time since 1950s.1 Wire media has

been revisited recently because of its profound use in the

metamaterial technology.2 It behaves as an artificial plasma

for waves propagating orthogonal to the wires, however, for

any other propagating direction it is characterized by strong

spatial dispersion (SD) effects (associated with a nonlocal

response of the matter, and it behaves very differently from a

material with indefinite anisotropic permittivity) at micro-

wave and low-terahertz frequencies, and even in the very

long-wavelength limit, i.e., in quasi-static.3

Despite these difficulties, it has been proposed that the

SD effects in the wire medium (WM) can be suppressed by

increasing the capacitance or the inductance of the wire (i.e.,

attaching large metallic plates or coating the wires with a

magnetic material).4 Based on these findings, it has been

shown that the SD effects can be significantly reduced by

attaching metallic patches to the WM,5,6 and a strong nega-

tive refraction at microwaves has been demonstrated in a

wire medium periodically loaded with metallic patches.7,8

On the other hand, in spite of the practical difficulties in find-

ing materials with desired magnetic response, recently it has

been shown that by loading the wires with lumped inductors

(increasing the inductance of the wires), the SD effects can

be significantly suppressed with the decrease in the plasma

frequency.9 In Ref. 9, all-angle negative refraction with a

high transmission has been demonstrated numerically by

considering a bi-layer mushroom structure with lumped

inductive loads. More recently, an alternative approach to

increase the inductance of the wire with reduced SD effects

has been presented by considering a racemic array of helical-

shaped wire medium.10

Here, we consider a bulk metamaterial formed by peri-

odically loading the wire medium with metallic patches and

lumped inductive loads (see Fig. 1), and further studying the

phenomenon of negative refraction, with a possibility to

demonstrate the effect of partial focusing of electromagnetic

radiation of transverse magnetic (TM) waves excited by a

magnetic line source. The motivation to consider this struc-

ture is because of its two important properties: all-angle neg-

ative refraction and high transmission,9 which are required

for the design of planar lenses which permit partial focusing.

It should be noted that the mechanism of partial focusing in

the proposed flat lens is somewhat similar to that achieved in

a flat slab of indefinite anisotropic material.11,12 Also, partial

focusing has been recently demonstrated with a crossed wire

mesh, even though it behaves as a material with a strong

nonlocal electromagnetic response.13,14

The analysis is carried out using the nonlocal homogeni-

zation model for the WM with the generalized additional

boundary conditions derived in a quasi-static approximation

by including arbitrary junctions with impedance insertions

(as lumped loads).9,15 Also, we model the combined

response of the structure using the continuous material

model which gives the qualitative description of the bulk ma-

terial as a local response. The proposed structure exhibits

high transmission and negative refraction below the plasma

frequency. The predictions of the homogenization model are

validated against the full-wave numerical results obtained

with a finite-element program HFSS.16

II. LOADED WIRE MEDIUM: HOMOGENIZATIONMODELS

The geometry of the multilayered mushroom configura-

tion considered in this work is shown in Fig. 1, with the TM-

polarized plane wave incidence. The structure consists of an

array of parallel conducting wires with radius r0 directed

along z in a host medium with relative permittivity eh. The

patch arrays are at the planes z ¼ 0;�h;�2h;…� L, and

the lumped inductive loads are inserted at the center of the

vias at the planes z ¼ �h=2;�3h=2; ::� ðLþ h=2Þ. The pe-

riod of the square patches is a and the gap is g. A time de-

pendence of the form ejxt is assumed and suppressed.

a)ckaipa@olemiss.edu.b)yakovlev@olemiss.edu.

0021-8979/2012/112(12)/124902/6/$30.00 VC 2012 American Institute of Physics112, 124902-1

JOURNAL OF APPLIED PHYSICS 112, 124902 (2012)

First, we model the bulk WM-type metamaterial (typical

geometry shown in Fig. 1) as a local uniaxial continuous ma-

terial, with no spatial dispersion. Within this framework, we

consider that the effect of the lumped loads and metallic

patches can be approximated by a uniform continuous load-

ing. Thus, the effective dielectric function includes the

response of the wires, metallic patches, and lumped loads.

Next, we take into account the SD effects in the wire me-

dium and the actual discreteness of lumped loads and metal-

lic patches. Within this alternative and more accurate

approach as outlined in Sec. II B, the effect of the lumped

loads and metallic patches is not incorporated in the effective

dielectric function of the material, but rather taken into

account through appropriate additional boundary conditions.

It should be noted that the generalized additional boundary

conditions are obtained based on a quasi-static model of wire

media that introduces two additional parameters: the wire

current and the additional potential at the wire surface.18 The

conditions at the junctions of wires with metallic patches and

at the connections of lumped loads to the wires are then for-

mulated in terms of these quantities.15 The obtained addi-

tional boundary conditions can be reformulated in terms of

the electric and magnetic field components tangential to the

surface, and have been applied successfully to different con-

figurations of mushroom-type structures (including the case

of vias loaded with impedance insertions) to study the reflec-

tion/transmission properties and negative refraction showing

a good agreement with full-wave numerical simulations.9,15

A. Continuous material model

Here, we model the combined response of the metallic

wires, patches, and the lumped inductive loadings in terms

of a continuous material model (local response) based on the

quasistatic approach.9,18 Following Ref. 9, the permittivity

dyadic of the uniaxial WM uniformly loaded with metallic

patches and impedance insertions can be expressed as

follows:

��ee0

¼ et��I t þ 1�

~k2

p

k20

!z0z0

!; (1)

where et ¼ 1þ 2ða� gÞ=phlog½cscðpg=2aÞ� is the transverse

permittivity19 for the patch arrays separated by a distance

h, ~kp ¼ kp=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Lload=ðhLwireÞ

pis the effective plasma

wavenumber, Lload is the value of the lumped inductance,

Lwire ¼ ðl0=2pÞlog½a2=ð4r0ða� r0ÞÞ� is the inductance per

unit length of the WM, and ��I t is the unit dyadic in the plane

orthogonal to z0 (a unit vector in the z-direction). This model

does not take into account the granularity of the structure

along z, i.e., the loading is assumed to be effectively continu-

ous along the wires. The transmission/reflection properties

based on the continuous material model can be obtained by

matching the tangential electric and magnetic fields at the

air-material interfaces. Since, the model does not take into

account the SD effects in the WM, it does not require any

additional boundary conditions. It should be noted that the

results of the continuous material model tend to be signifi-

cantly less accurate (results not reported here for sake of

brevity), particularly near the plasma frequency of the struc-

ture, however, they give a qualitative description of the

response. This simple model is valid, provided that the SD

effects are significantly suppressed in the WM. Nevertheless,

in this work this is the case because by employing inductive

loads, the bulk metamaterial behaves predominantly as a ma-

terial with local indefinite response.9 Also, this model is

accurate only for a bulk medium.

B. Discrete loading model

In the analysis based on the discrete loading model, the

patch arrays are treated as impedance sheets,17 and the WM

slab is modelled as a uniaxial continuous material character-

ized by a spatially dispersive effective dielectric function

along the direction of wires: ezz ¼ ð1� k2p=ðk2

0 � k2z ÞÞ, where

kp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2p=a2Þ=log½a2=4r0ða� r0Þ�

pis the plasma wave-

number as defined in Ref. 18, k0 ¼ x=c is the free-space

wavenumber, and kz is the z-component of the wave vector ~kinside the material.

The reflection/transmission properties of the multilayered

mushroom configuration can be obtained in the same manner

as presented for the bi-layer mushroom structure,9 by using the

transfer matrix approach. Following Ref. 9, in addition to the

two-sided impedance boundary conditions at the patch interfa-

ces and additional boundary conditions at the connection of the

metallic wires to the metallic patches (taking into account the

nonlocal response in the WM), the discontinuities in the

microscopic wire current distribution I(z), at the connections

of the wires through lumped loads (at the planes

z ¼ z0 ¼ �h=2;�3h=2; ::� ðLþ h=2Þ) are taken into account

by the following generalized additional boundary conditions:15

I1ðzÞjz¼zþ0¼ I2ðzÞjz¼z�

0; (2)

dI2ðzÞdz jz¼z�

0

� dI1ðzÞdz jz¼zþ

0

¼ �jxCwireZloadI1ðzÞjz¼zþ0; (3)

FIG. 1. 3-D geometry of a wire medium periodically loaded with metallic

patches and lumped loads at the center (along the direction of the wires)

excited by an obliquely incident TM-polarized plane wave.

124902-2 C. S. R. Kaipa and A. B. Yakovlev J. Appl. Phys. 112, 124902 (2012)

where I1ðzÞ and I2ðzÞ correspond to the microscopic wire

currents in the medium above and below the lumped

inductive load, respectively, at the plane z ¼ z0. Cwire ¼2pe0=log½a2=ð4r0ða� r0ÞÞ� is the capacitance per unit length

of the wire medium18 and Zload ¼ jxLload is the impedance

of the lumped inductance (Lload).

In the Sec. III, the predictions of the analytical models

are presented along with the full-wave numerical results.

III. RESULTS AND DISCUSSION

First, the analysis of the bulk metamaterial is carried out

using the continuous material model (described in Sec. II A).

Even though the results of the continuous material model are

quantitatively poor, we explore its advantage because of its

qualitative understanding and simple modeling when com-

pared to the nonlocal model. In order to study the possibility

of partial focusing by a flat lens, one requires a thick slab of

the metamaterial. Here, we consider a twenty-five-layer

mushroom structure (L¼ 25a) formed by twenty six identical

patch arrays and twenty five identical inductively loaded

WM slabs. The geometrical parameters are as follows:

a¼ 2 mm, g¼ 0.2 mm, h¼ 2 mm, r0 ¼ 0:05 mm; eh ¼ 1, and

the inductive load Lload ¼ 5 nH. Now, we suppose that a

magnetic line source infinitely extended along the y-direction

(a y-polarized magnetic line source excites a TMx;z field) is

placed at a distance d¼ 0.5 L above the mushroom slab. We

operate at the frequency of 10.85 GHz, where the effective

permittivity components calculated using Eq. (1) are: exx ¼eyy ¼ 1:51 and ezz � �1. Assuming that the metamaterial

slab is unbounded along the lateral directions x and y, it is

straightforward to show that the magnetic field in the three

regions of space can be written in terms of the Sommerfeld-

type integrals. In Fig. 2, the magnetic-field profiles jHyj are

depicted in the x-z plane for the twenty-five-layered

mushroom slab. It can be clearly seen that there is an intense

partial focus of the magnetic field inside and also below the

mushroom lens, thus showing that a flat slab of mushroom

with inductive loadings can indeed redirect the electromag-

netic radiation of a TM-polarized line source to a narrow

spot at the focal plane. An independent verification of the

observed phenomena is performed by simulating a finite

block of anisotropic material (with the parameters calculated

from Eq. (1)) using full-wave numerical solver HFSS, result-

ing in the same behavior (results not reported here for sake

of brevity). The calculated half power beam-width (HPBW)

at the image plane is �k=2 (diffraction limit).

To further confirm these findings, we study the transmis-

sion properties and characterize the negative refraction based

on the nonlocal (discrete loading) homogenization model (as

described in Sec. II B) and numerically demonstrate the partial

focusing effect using full-wave solver HFSS. Due to computa-

tional limitations in modeling a realistic structure, we limit

our analysis to a seven-layer mushroom slab. Figure 3 shows

the nonlocal homogenization results for the transmission mag-

nitude and phase as a function of frequency calculated for the

seven-layer mushroom slab illuminated by a TM-polarized

plane wave incident at angle of 45�. The geometrical parame-

ters are the same as those used for the calculations of Fig. 2. It

is seen that the homogenization results are in good agreement

with the full-wave HFSS results, except for a small shift in

the plasma frequency. It can be observed from Fig. 3, that the

structure exhibits a high transmission which is fairly a con-

stant except in the close vicinity of the plasma frequency.

This is because the air-filled configuration is better matched to

free space, and also the patch arrays at low-frequencies are

highly transparent and behave predominantly as low-pass fil-

ters with a flat frequency response.20

Next, we calculate the transmission angle ht as a func-

tion of the incidence angle hi of the impinging TM-polarized

plane wave. We characterize the transmission angle based on

the analysis of variation in the phase of Tðx; kxÞ (transmis-

sion coefficient for a plane wave with the transverse wave-

number kx) of the material slab with the incident angle hi.

Specifically, it was shown in Ref. 21 that for an arbitrary ma-

terial slab excited by a quasi-plane wave, apart from the

transmission magnitude, the field profile at the output plane

differs from the same at the input plane by a spatial shift D,

given by D ¼ d/=dkx, where / ¼ argðTÞ. The transmission

angle can be obtained as ht ¼ tan�1ðD=LÞ (L is the thickness

of the planar material slab). Fig. 4(a) shows the calculated ht

as a function of hi at the frequencies of 9.9 GHz and

11.5 GHz. The frequencies were selected based on the para-

metric study, such that ht is reasonably a linear function of hi

in the range 0 < hi < 45�. It should be noted that in the case

of Pendry’s lens22 (formed with e ¼ �1 and l ¼ �1),

jhtj ¼ jhij, and consequently the thickness L¼ 2d provides

perfect focusing. However, in the considered indefinite mate-

rial, the angle of transmission ht is a nonlinear function of

the incidence angle hi. The ray-tracing diagram is depicted

in Fig. 4(b) showing the path of the rays inside and

outside the slab for d¼ 0.5 L. It is seen that the rays coming

from the line source (located above the slab) are partially

refocused inside the slab, and also below the slab at the same

distance d.

FIG. 2. Continuous material model results of the amplitude of the magnetic-

field distribution jHyj, showing the partial focusing effect by the mushroom

lens. The frequency of operation is 10.85 GHz, the magnetic line source is

located at a distance d ¼ 0:5L � 0:90k0 from the upper interface of the slab,

and L ¼ 25a � 1:81k0. The blue solid lines represent the interfaces of the

mushroom lens.

124902-3 C. S. R. Kaipa and A. B. Yakovlev J. Appl. Phys. 112, 124902 (2012)

Based on the predictions of the homogenization model,

the performance of the proposed mushroom-type lens is stud-

ied using the commercial electromagnetic simulator HFSS.16

The magnetic line source is created in HFSS by considering

a voltage source excited in the form of a square loop (in the

x-z plane) and considering perfect magnetic conductor

boundary conditions (Ht ¼ 0) at the planes y¼ 0, a. In the

full-wave simulations, the load is inserted in the wires

through a gap of 0.1 mm. The artificial material slab was

assumed periodic along the y-direction and finite along the

x-direction. The width of the slab was taken to be equal to

Wx ¼ 25a along the x-direction, with a¼ 2 mm being the pe-

riod of the unit cell. The metallic components (wires and

patches) are modelled as copper metal with the bulk conduc-

tivity rCu ¼ 5:8� 107 S=m, taking into account the effect of

the ohmic losses.

Fig. 5(a) shows the snapshot in time of the magnetic

field Hy at t¼ 0 in the x-z plane, calculated at 10 GHz. It is

assumed that the magnetic line source is placed at a distance

d ¼ 7 mm ¼ 0:23k0 (k0 corresponds to the free-space wave-

length at the operating frequency of 10 GHz) above the

mushroom slab, and the image plane is located at the same

distance (d) from the lower interface of the slab. It can be

clearly seen that there is an intense partial focus of the mag-

netic field inside the mushroom slab, and also below the

lens. However, the focal point at the image plane partially

overlaps with the lower interface of the slab. This may be

due to the fact that the transmission angle ht is fairly a con-

stant for the incidence angles hi > 35� (see Fig. 4(a)), which

results in a focusing closer to the structure. Also in Fig. 5(b),

we show the calculated square-normalized magnetic-field

profile at the image plane. The magnetic field is calculated

with HFSS, along a line parallel to the slab at the image

plane. The calculated HPBW is near 0:4k0, close to the dif-

fraction limit value (k0=2).

Next, we study the focusing effect of the mushroom slab

at the frequency of 12 GHz. The snapshot of the calculated

magnetic-field profile is depicted in Fig. 6(a), clearly show-

ing an intense partial focus below the lower interface of the

FIG. 4. (a) Homogenization results of the transmission angle ht as a function

of incidence angle hi for the seven-layer mushroom structure calculated at

the frequencies of 9.9 GHz and 11.5 GHz and (b) ray-tracing diagram show-

ing that the structured material refocuses the rays coming from a line source

both inside and outside of the slab. The source is placed at a distance

d¼ 0.5 L from the front interface and the thickness of the slab is L¼ 7 a.

The frequency of operation is 9.9 GHz.

FIG. 3. Transmission characteristics for the seven-layer mushroom structure

excited by a TM-polarized plane wave incident at 45� as a function of fre-

quency. (a) Magnitude of the transmission coefficient. (b) Phase of the trans-

mission coefficient. The solid lines represent the results of the nonlocal

homogenization model and the symbols correspond to the full-wave HFSS

results.

124902-4 C. S. R. Kaipa and A. B. Yakovlev J. Appl. Phys. 112, 124902 (2012)

slab. The square-normalized amplitude of the magnetic-field

profile as a function of x=k0 is shown in Fig. 6(b). The calcu-

lated HPBW is 0:38k0, and is smaller when compared to the

case of 10 GHz. This is because in the case of 10 GHz, the

focus point coincides with the lower interface of the slab and

is not exactly at the image plane, where we calculate the field

profile.

IV. CONCLUSION

We have shown that it is possible to design bulk meta-

material formed by periodically loading wire medium with

metallic patches and lumped inductive loads, which exhibits

a high transmission and negative refraction over a wide fre-

quency band at microwaves. We demonstrate that the loaded

wire medium slab (flat lens) focuses electromagnetic radia-

tion both inside and below the slab. The calculated HPBW

of the focus point at the image plane is 0:4k0. The response

can be accurately predicted by the effective medium model.

ACKNOWLEDGMENTS

The authors would like to acknowledge the helpful

discussions with Mario G. Silveirinha and Stanislav I.

Maslovski.

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(a) (b)

FIG. 5. (a) Snapshot in time of the magnetic field Hy with the magnetic line source placed at a distance d ¼ 0:23k0 from the upper interface of the structure.

(b) Square-normalized amplitude of Hy calculated along a line parallel to the slab at the image plane. The frequency of operation is 10 GHz.

(a) (b)

FIG. 6. (a) Snapshot in time of the magnetic field Hy with the magnetic line source placed at a distance d ¼ 0:28k0 from the upper interface of the structure.

(b) Square-normalized amplitude of Hy calculated along a line parallel to the slab at the image plane. The frequency of operation is 12 GHz.

124902-5 C. S. R. Kaipa and A. B. Yakovlev J. Appl. Phys. 112, 124902 (2012)

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124902-6 C. S. R. Kaipa and A. B. Yakovlev J. Appl. Phys. 112, 124902 (2012)