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:
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

,2

), Region 3 is a metamaterial
((

3

 ),

3

 ), Region 4 is a regular dielectric
(4

,4

). 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]:


1 11
1ˆ
eee
x zz
ik xt
ik zik z
A
By
 

E (1)
To find the corresponding magnetic field 1
H
, we start
with Maxwell’s equation [15]:
1
1t



B
E (2)
substituting 111


B
H and solving for 1
H
yield:



11
1 11
111
1
11
1ˆ
ee
ˆ eee
zz
x zz
ik zikz
xx
ik xt
ik zik z
zz
Ak Bk z
Ak Bkx









 
H
(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:



2 22
2ˆ
ex
zz
ik xt
ik zik z
Ce Dey
 

E (4)



22
2 22
222
2
22
1ˆ
ee
ˆ eee
zz
x zz
ik zik z
xx
ik xt
ik zikz
zz
Ck Dk z
Ck Dkx









 
H
(5)



3 33
3ˆ
eee
x zz
ik xt
ik zik z
F
Gy
 

E (6)



33
3 33
333
3
33
1ˆ
ee
ˆ eee
zz
x zz
ik zik z
xx
ik xt
ik zik z
zz
Fk Gkz
Fk Gkx









 
H
(7)

4 4
4ˆ
ee
x zik xt
ik z
J
y
 
E (8)


4
44
44 4
4
1ˆ
ˆ eee
x
zz
ik xt
ik zik z
xz
Jkz Jkx



H (9)
where ii
kn c


is the wave vector inside the mate-
rial and ii
i
oo
n




 is the refractive index of it.
Matching the boundary conditions for
E
and
H
fields at each layer interface, that is at z = 0, 12

E
E
and H1 = H2, at z = a 23

E
E and 23 
H
H, and at z
= a + d, E3 = E4 and 34

H
H. This yields the follow-
ing equations [10,12-15]:
A + B = C + D (10)
 
12
12
zz
kk
A
BCD

  (11)
33 22
ee ee
z
z zz
ik aik a ik aik a
CD FG
 
 (12)
 
33 22
3 2
23
ee ee
zz zz ik aik a
ik aik az
zk
kCDFG



  (13)
 
334
ee e
zzz
ikadikadika d
FG J
 
 (14)
 


33 4
34
34
ee e
zz z
ikadikadika d
zz
kk
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 zz
RkkT

 (18)

M. F. Ubeid et al. / Natural Science 3 (2011) 328-333 330
where:
222
1sin iz i
knn
c


 (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]:

2
322
1eep
eo e
F
i




  (20)

2
322
1mmp
mo m
F
i




  (21)
where ep

and mp

are the electric and magnetic
plasma frequencies, eo

and mo

are the electric and
magnetic resonance frequencies. e
F
and m
F
are the
scaling filling parameters.
We have used the following parameters in [15]:
2π10.95 GHz
mp

, 2π10.1 GHz
mo

, 0.26
m
F

,
2π13.3 GHz
ep

,2π10.3 GHz
eo

, 0.37
e
F, with
no loss case i.e. 0
em

 . In this case, the frequency
range in which

3

 and

3

 are negative ex-
tends from 10.3 up to 11.4 GHz. The obtained values of

3

 ,

3

 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

on

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).
14
nn

(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 nn

)
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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