Effect of negative permittivity and permeability on the transmission of electromagnetic waves through a structure containing left-handed material

doi:10.4236/ns.2011.34043

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Vol.3, No.4, 328-333 (2011) Natural Science

http://dx.doi.org/10.4236/ns.2011.34043

Effect of negative permittivity and permeability on the

transmission of electromagnetic waves through a

structure containing left-handed material

Muin F. Ubeid1*, Mohammed M. Shabat1, Mohammed O. Sid-Ahmed2

1Department of Physics, Faculty of Science, Islamic University of Gaza, Gaza, Palestine;

*Corresponding Author: [email protected]

2Department of Physics, Faculty of Science, Sudan University of Science and Technology, Khartoum, The Republic of The Sudan

Received 5 March 2011; revised 23 March 2011; accepted 27 March 2011.

ABSTRACT

We investigate the characteristics of electro-

magnetic wave reflection and transmission by

multilayered structures consisting of a pair of

left-handed material (LHM) and dielectric slabs

inserted between two semi-infinite dielectric

media. The theoretical aspect is based on

Maxwell's equations and matching the boundary

conditions for the electric and magnetic fields of

the incident waves at each layer interface. We

calculate the reflected and transmitted powers

of the multilayered structure taking into account

the widths of the slabs and the frequency de-

pendence of permittivity and permeability of the

LHM. The obtained results satisfy the law of

conservation of energy. We show that if the

semi-infinite dielectric media have the same

refractive index and the slabs have the same

width, then the reflected (and transmitted) pow-

ers can be minimized (and maximized) and the

powers-frequency curves show no ripple. On

the other hand if the semi-infinite dielectric me-

dia have different values of refractive indices

and the slabs have different widths, then under

certain conditions the situation of minimum and

maximum values of the mentioned powers will

be reversed.

Keywords: Electromagnetic Waves; Left-Handed

Materials; Frequency; Reflected and Transmitted

Powers

1. INTRODUCTION

Metamaterials (sometimes termed left-handed materi-

als (LHMs)) are materials whose permittivity



and

permeability



are both negative and consequently

have negative index of refraction. These materials are

artificial and theoretically discussed first by Veselago [1]

over 40 years ago. The first realization of such materials,

consisting of split-ring resenators (SRRs) and continuous

wires, was first introduced by Pendry [2,3]. Regular ma-

terials are materials whose



and



are both positive

and termed right handed materials (RHMs). R. A. Shelby

et al. [4] have studied negative refraction in LHMs. I. V.

Shadrivov [5] has investigated nonlinear guided waves

in LHMs. N. Garcia et al. [6] have shown that LHMs

don’t make a perfect lens. Kong [7] has provided a gen-

eral formulation for the electromagnetic wave interaction

with stratified metamaterial structures. M. M. Shabat et

al [8] have discussed Nonlinear TE surface waves in a

left-handed material and magnetic super lattice wave-

guide structure. I. Kourakis et al. [9] have investigated a

nonlinear propagation of electromagnetic waves in nega-

tive–refraction index LHM. H. Cory et al. [10] and C.

Sabah et al. [11] have estimated high reflection coatings

of multilayered structure. Oraizi et al. [12] have obtained

a zero reflection from multilayered metamaterial struc-

tures.

In this paper we consider a structure consisting of

LHM and dielectric slabs inserted between two semi-

infinite dielectric media. A plane polarized wave is

obliquely incident on it. We use Maxwell’s equations

and match the boundary conditions for the electric and

magnetic fields of the incident waves at each layer in-

terface. Then we solve the obtained equations for the

unknown parameters to calculate the reflection and

transmission coefficients. We take into account the fre-

quency dependence of permittivity and permeability of

the LHM (in contradict with [10,11]), widths of the slabs,

refractive indices of the media and angle of incidence of

the incident waves. Maximum and minimum transmitted

(minimum and maximum reflected) powers of the con-

sidered structure are proposed. The numerical results are

M. F. Ubeid et al. / Natural Science 3 (2011) 328-333

329

in agreement with the law of conservation of energy

given by [10,13]. It is found that the numerical results of

Figure 3 (the case a ≠ b) is similar to Figure 4(b) ob-

tained by [14], this is another evidence for validity of the

performed computations.

The paper is organized as follows: our theory is for-

mulated in section 2. Numerical results and applications

are described in section 3. Our conclusions are presented

in section 4.

2. THEORY

Consider four regions each with permittivity i



and

permeability i



, where i represents the region order.

Region 1 is a vacuum (1o



,1o



), Region 2 is a

regular dielectric (2





), Region 3 is a metamaterial

((













), Region 4 is a regular dielectric





). A polarized plane wave in Region 1 incident

on the plane z = 0 at some angle



relative to the nor-

mal to the boundary (see Figure 1).

The electric field of the incident wave in Region 1 can

be written as [7-10]:



1ˆ

eee

ik xt

ik zik z





E (1)

To find the corresponding magnetic field 1

, we start

with Maxwell’s equation [15]:







E (2)

substituting 111





H and solving for 1

yield:



111

1ˆ

eee

ik zikz

ik xt

ik zik z

Ak Bk z

Ak Bkx















 

(3)

Figure 1. Wave propagation through a structure consisting of a

pair of dielectric and metamaterial embedded between two die-

lectric semi-infinite media.

The electric and magnetic fields in Regions 2, 3 and 4

can be written in the same manner as follows:







2ˆ

ik xt

ik zik z

Ce Dey





E (4)



222

1ˆ

eee

ik zik z

ik xt

ik zikz

Ck Dk z

Ck Dkx















 

(5)







3ˆ

eee

ik xt

ik zik z





E (6)



333

1ˆ

eee

ik zik z

ik xt

ik zik z

Fk Gkz

Fk Gkx















 

(7)



4ˆ

zik xt

ik z





E (8)



44 4

1ˆ

eee

ik xt

ik zik z

Jkz Jkx







H (9)

where ii

kn c





is the wave vector inside the mate-

rial and ii









 is the refractive index of it.

Matching the boundary conditions for

and

fields at each layer interface, that is at z = 0, 12



and H1 = H2, at z = a 23



E and 23



H, and at z

= a + d, E3 = E4 and 34



H. This yields the follow-

ing equations [10,12-15]:

A + B = C + D (10)

 

BCD



  (11)

ee ee

ik aik a

CD FG



 (12)



ee ee

zz ik aik a

ik aik az

kCDFG





  (13)

 

334

ee e

zzz

ikadikadika d

FG J

 

 (14)

 



33 4

ee e

zz z

ikadikadika d

FG J





 

 (15)

Letting A = 1 and solving these six equations for the

unknown parameters enable us to calculate the reflection

and transmission coefficients B and J respectively

[10,15]. The reflected power R equal to the reflection

coefficient B times its complex conjugate and the trans-

mitted power T equal to the transmission coefficient J

times its complex conjugate [10,15], leading to;

RBB



(16)

TJJ (17)

The law of conservation of energy is given by [10,

13]:





41 1

RkkT



 (18)

M. F. Ubeid et al. / Natural Science 3 (2011) 328-333

330

where:

222

1sin

iz i

knn





 (19)

3. NUMERICAL RESULTS AND

APPLICATIONS

For the metamaterial in region 3 we will employ a

non-dispersive metamaterial with



and



given by

[2,3,12,15-17]:



322

1eep

eo e









  (20)



322

1mmp

mo m









  (21)

where ep



and mp



are the electric and magnetic

plasma frequencies, eo



and mo



are the electric and

magnetic resonance frequencies. e

and m

are the

scaling filling parameters.

We have used the following parameters in [15]:

2π10.95 GHz



, 2π10.1 GHz



, 0.26



2π13.3 GHz



,2π10.3 GHz



, 0.37

F, with

no loss case i.e. 0



. In this case, the frequency

range in which







and







are negative ex-

tends from 10.3 up to 11.4 GHz. The obtained values of













in addition to iz

k are used in (10-15).

These equations are solved for the parameters B and J.

Then the reflected and transmitted powers R and T can

be calculated. The transmitted power is plotted as a

function of



under different conditions as follows:

The dependence of 3





for the metamaterial

in Region 3 is taken into account [2-5,15-17], it is con-

sidered by [10, 11] to be 3o





. The difference be-

tween the two cases can be noticed from Figure 2.

The thickness of the slabs is taken to be the same a =

d, 23

nn, 14

nn,



is kept constant. In this case

the structure’s reflected and transmitted powers variation

with frequency are smooth and show no ripples as shown

in Figure 3. To check the validity of computations for

the case a ≠ b as an example, let ω = 2π11 GHz, then: μ3

= –0.641 635 071, ε3 = –3.389 624 4113, k4z/k1z = 1, B =

–0.570 959 7786 + 2.958 090 86i, J = –0.743 521 1938 +

0.184 524 7865i, R = 0.413 126 8376, T = 0.586 873 1625.

In this case the left hand side of (18) is equal to 0.999

999 99 which verifies the law of conservation of energy.

It can be realized from Figure 3 that this law is satisfied

by the performed computations. The same procedure can

be applied to other computations in this paper.

Low reflected and high transmitted powers can be

achieved if 14

nn, 23

nn and a = d [10]. In this

case R = 0 (minimum) and T = 1 (maximum) for any

frequency and for any angle of incidence (Figure 4). If

Figure 2. Transmitted power variation with frequency. Two

cases are taken into account: μ3 is a function of ω, μ3 is a con-

stant (–μ0).



(and a = d, 23

nn), then both R and T de-

pend on the values of the refractive indices of the initial

and final media 1

n and 4

n, and on the angle of inci-

dence



see Figures 5 and 6.

High reflected and Low transmitted powers: in order

to maximize R and minimize T, one has to choose a pair

of adjacent dielectric and metamaterial slabs with highly

contrasted refractive indices (12

nn,23

nn, 34



)

and 23πo

na ndc





 [10], where o



is the central

frequency (o



= 10.9 GHz). In this case by a judicious

combination of metamaterial and dielectric slabs, a

high-reflected and low-transmitted powers are achieved,

for which the dependence of R and T on frequency and

on the angle of incidence is consequently diminished

(Figure 7). Note that the maximum value of T is 0.09 at

0o. This value is smaller by a factor of 11 than that ob-

tained in Figure 4. Figure 7 is different from that ob-

tained by [10,11] in the fact that, they had used a LHM

with properties invariant with frequency. In our paper,

the properties of the LHM in Region 3 depends on fre-

quency (this can be realized from Eqs.20 and 21).

4. CONCLUSIONS

The propagation of electromagnetic waves through

multilayered structures consisting of a pair of LHM and

dielectric slabs inserted between semi-infinite dielectric

media has been studied. The followed method has been

based on Maxwell's equations and matching the bound-

ary conditions for the electric and magnetic fields at

each interface layer. The frequency dependence of



and



of the LHM has been taken into account. The

reflected and transmitted powers have been calculated

numerically. The dependence of them on various pa-

rameters has been studied. Low and high transmitted

powers have been achieved for any frequency and for

any angle of incidence. The law of conservation of energy

M. F. Ubeid et al. / Natural Science 3 (2011) 328-333

331

(a) (b)

Figure 3. The reflected and Transmitted powers variation with frequency when n1 = n4, n2 = |n3| and θ is kept constant for two cases

with respect to the widths a and d of the slabs: a = d, a ≠ d.

(a) (b)

Figure 4. The reflected and transmitted powers as a function of frequency when n2 = |n3|, n1 = n4 and a = d for various angle of inci-

dence: θ = 0˚, θ = 30˚, θ = 50˚.

(a) (b)

Figure 5. The reflected and transmitted powers against frequency when n1 ≠ n4, n2 = |n3| and a = d for various angle of incidences: θ

= 0˚, θ = 30˚, θ = 50˚, θ = 70˚, θ = 90˚.

has been satisfied by the obtained results. The depend-

ence of propagation on the given parameters gives rise to

possibilities of tuning the transmitted power for applica-

tions in microwave, antenna radome, millimeter wave,

M. F. Ubeid et al. / Natural Science 3 (2011) 328-333

332

(a) (b)

Figure 6. The reflected and transmitted powers versus frequency when n1 ≠ n4, n2 = |n3| and a = d, and the angle of incidence θ = 30˚

(kept constant) for different value of n4: n4= 1, n4= 1.66, n4 = 2.25.

(a) (b)

Figure 7. The reflected and transmitted powers as a function of frequency where the condition (n1 < n2, n2 > n3, n3 < n4) is satisfied

for various angle of incidences: θ = 0˚, θ = 30˚, θ = 50˚, θ = 70˚.

and optical devices.

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