This paper presents asymmetric three-layer slab waveguide structure in which all layers are considered
left-handed materials (LHMs) with negative electric permittivity and magnetic permeability. The
electric field profile is investigated in details with all parameters of the waveguide structure. Many interesting
features are observed such as the existence of the fundamental mode and the transition from the
fundamental mode to the first guided mode with increasing the imaginary part of the guiding layer permittivity
or permeability.
When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/3598

Elwasife, Kh.Y.

Taya, S.A.

SocioEconomic Challenges Journal (Sumy State University)

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ 
Vol. 6 No 2, 02007(5pp) (2014) Том 6 № 2, 02007(5cc) (2014) 
 
 
2077-6772/2014/6(2)02007(5) 02007-1  2014 Sumy State University 
Field Profile of Asymmetric Slab Waveguide Structure with LHM Layers 
 
S.A. Taya*, Kh.Y. Elwasife†  
 
Physics Department, Islamic University of Gaza, Palestinian Authority 
 
(Received 16 March 2014; published online 20 June 2014) 
 
This paper presents asymmetric three-layer slab waveguide structure in which all layers are consid-
ered left-handed materials (LHMs) with negative electric permittivity and magnetic permeability. The 
electric field profile is investigated in details with all parameters of the waveguide structure. Many inter-
esting features are observed such as the existence of the fundamental mode and the transition from the 
fundamental mode to the first guided mode with increasing the imaginary part of the guiding layer permit-
tivity or permeability. 
 
Keywords: Slab waveguides, Guided modes, Left-handed materials.  
 
 PACS numbers: 42.65.Wi, 42.65.Tg, 42.25.Bs 
 
 
                                                                
*
 staya@iugaza.edu.ps 
†
 kelwasife@iugaza.edu.ps 
1. INTRODUCTION 
 
Several groups have succeeded in fabricating left-
handed media and it is now possible to employ left-handed 
electromagnetism in a large range. In recent years, the 
topic of metamaterials, i.e. artificial materials synthesized 
by embedding specific inclusions in host media, has in-
creasingly received a renewed attention due to the inter-
est in man-made complex materials that may possess 
negative permittivity (ε) and permeability (μ) in a certain 
frequency band. In 1968, Veselago postulated theoretically 
a material in which both ε and μ were assumed to have 
negative real values and he analyzed plane wave propaga-
tion in such a medium [1]. These materials were called 
left-handed metamaterials (LHMs). Since then an inten-
sive research has been carried out to understand the 
properties of electromagnetic waves in these media [2-20]. 
I. V. Shadrivov analyzed nonlinear guided waves in a pla-
nar waveguide made of a LHM surrounded by a Kerr-like 
nonlinear dielectric and predicted that such a waveguide 
can support fast and slow symmetric and antisymmetric 
nonlinear modes [2]. Moreover, he studied the symmetry 
breaking bifurcation and asymmetric modes in such a 
symmetric structure and showed that the modes can be 
either forward or backward [2]. Characteristics of surface 
wave modes on a grounded slab of LHM were investigated 
[3]. It was shown that, unlike a slab with positive parame-
ters, the dominant mode can have evanescent fields on 
both sides of the interface between the slab and the sur-
rounding air. The guided and surface modes in an LHM 
asymmetric slab waveguide were studied by introducing 
three normalized parameters [4-8]. The guided dispersion 
characteristics of the fundamental symmetric and asym-
metric modes of surface waves along single- and double-
negative indexed slab waveguides were investigated and a 
comparative analysis was made when varying the single- 
and double-negative permittivity and permeability [9]. 
Asymmetric waveguide structure with a LHM guiding 
layer surrounded by a linear dielectric substrate and a 
gyromagnetic ferrite cladding has been investigated [10]. 
Guided modes supported by an asymmetric three-layer 
metal-clad waveguide structure were presented when the 
guiding layer is LHM [11]. Multilayer planar waveguide 
structures comprising LHMs have been explored for both 
TM and TE polarizations [12-14]. TE surface modes guid-
ed by a single interface and a slab waveguide containing 
LHMs were analyzed by using a ray picture [15]. The 
transfer function of the discretized perfect lens in finite-
difference time-domain and transfer matrix method simu-
lations was investigated [16]. Waveguide structures com-
prising LHMs have been strong candidates for many opto-
electronics applications. One of the most interesting appli-
cations of LHM waveguide structures is optical sensing 
[17-20] in which any change in the refractive index of the 
analyte is detected.  
In this paper asymmetric three-layer planar wave-
guide structure is studied for both TE and TM modes. 
All layers are considered to be LHMs. The field profile 
is investigated in details with all parameters of the 
LHMs.  
 
2. THEORY 
 
We suppose an asymmetric three-layer slab waveguide 
structure. The substrate, guiding film, and cladding are 
assumed to be LHMs with εi  0 and I  0 (i  1, 2, 3). The 
waves are considered to propagate along z-direction and x-
axis is perpendicular to the plane separating the structure 
layers as shown in Fig. 1. 
 
d
z
x
LHM film ε1, µ1
LHM cald ε2, µ2
LHM substrate ε3, µ3
 
 
Fig. 1 – Cross-sectional view of asymmetric LHM slab waveguide 
 
 S.A. TAYA, KH.Y. ELWASIFE J. NANO- ELECTRON. PHYS. 6, 02007 (2014) 
 
 
02007-2 
Since the waveguide is homogeneous along z -axis, 
solutions to the wave equation can have the form  
 
 ˆ( , , ) ( )exp( ( ))y yE x z t E x j t z a   , (1) 
 
 ˆ ˆ( , , ) [ ( ) ( ) ]exp( ( ))x x z zH x z t H x a H x a j t z    , (2) 
 
where  is the angular frequency and β is the propaga-
tion constant. The wave vector components h, q2, and q3 
are defined as 2 20 1 1h k     , 
2 2
2 0 2 2q k    , 
2 2
3 0 3 3q k    . 
The y-component of the electric field for TE polari-
zation is given by  
 
 2
2
( sin( ) cos( )) 0< <
( ) exp( ) >d
exp( ) <0
y
A hx B hx x d
E x C q x x
D q x x


 


 (3) 
 
The magnetic field is given by 
 
 
E
H
j



, (4) 
 
resulting in  
 
 ( )
y
z
Ej
H x
x



 (5) 
 
The boundary conditions require that the tangential 
components of both E  and H  are continuous across 
the boundaries. The boundary condition equation for 
the asymmetric modes is given by 
 
 
1 3 1 2
3 2
1 3 1 2
3 2
tan( )
1
q q
h h
hd
q q
h h
 
 
 
 



 (6) 
 
To introduce the mode order (m), Eq. (6) can be written as 
 
 1 11 3 1 2
3 2
tan tan
q q
hd m
h h
 

 
         
  
. (7) 
 
The right –hand side (RHS) of Eq. (6) can be written as 
a function of (hd). We first assume 2q and 3q to have 
the forms 
 
 2 22 0 1 1 2 2( )q k h      , (8) 
 
 2 23 0 1 1 3 3( )q k h       (9) 
 
with these notation for 1q  and 3q , the RHS of Eq. (6) 
can be written as  
 
 
2 2 2 232
1 1
2 2 2 2 22 3
2
1
( ) ( )
( )
( ) ( ) ( )
s hd p hd
F hd hd
hd s hd p hd

 
 

  

  
(10) 
 
where  0 1 1 2 2s k d       and  0 1 1 3 3p k d      . 
When propagation constant  exceeds a critical val-
ue, the wave number h becomes purely imaginary. 
For TM polarization, the boundary condition equa-
tion is given by 
 
 
1 3 1 2
3 2
1 3 1 2
3 2
tan( )
1
q q
h h
hd
q q
h h
 
 
 
 



 (11) 
 
3. RESULTS AND DISCUSSION 
 
Figure 1 shows the slab waveguide geometry under 
consideration. In the analysis below, the frequency of 
electromagnetic waves is assumed to be 4.6GHz. Oscil-
lating mode can exist in the proposed structure for 
2 2 2
0 1 1 0 3 3> >k k     . Graphical method can be used to 
determine the values of hd for possible guided modes. 
Tan (hd) given by Eq. (6) and F(hd) given by Eq.(10) 
are plotted versus hd in Fig. 2. The intersection points 
give the values of hd at which the guided modes exist. 
It is clear from the figure that the proposed structure 
can support guided modes. Transverse profiles of the 
TE0 mode for different values of the operating frequen-
cy are plotted in Fig. 3 for s  f  c  – 1 + 0.001i, 
εs  – 4.7 + 0.001i, εf  – 5.7 + 0.001i, and εc  
 – 3.7 + 0.001i. The guiding layer thickness is assumed 
to be 6 m. As can be seen from the figure, the electric 
field oscillates in the guiding layer and is evanescent in 
the surrounding media. The two tails of the evanescent 
are asymmetric in the substrate and cladding due to 
the asymmetric structure assumed. Increasing the an-
gular frequency of the guided wave from 4.6 GHz to 
5 GHz does not have a considerable effect on the field 
profile. It merely enhances the peak of the oscillating 
waves in the guiding layer. Moreover, it causes a barely 
detectable reduction in the evanescent field in the sur-
rounding media. It is worth to mention that the wave-
guide structure considered in this work can accommo-
date the fundamental mode. The absence of the funda-
mental mode was reported in many works in literature 
[7] when a LHM guiding layer is surrounded by two 
linear dielectric media. Our investigation here of a 
LHM guiding layer surrounded by two LHMs empha-
sizes that this structure can support any mode includ-
ing the fundamental one. Transverse profiles of the 
fundamental mode for different values of the permittiv-
ity (left) and permeability (right) of the film are shown 
in Fig. 4. As can be seen from the figure, increasing the 
absolute value of Re(εf) or Re(f) reduces the evanescent 
field in the substrate and cladding layers. Moreover, it 
leads to an enhancement of the oscillating wave peak in 
the guiding layer. This property is very similar to that 
of conventional guided modes in the well-known dielec-
tric waveguides in which the increasing of the guiding 
layer refractive index enhances the confinement of the 
wave and reduces the wave tails in the surrounding 
media. The distributions of electric field amplitude in 
the waveguide for different values of the substrate 
permittivity (left) and permeability (right) are shown in 
Fig. 5. The most important feature that can be ob-
 FIELD PROFILE OF ASYMMETRIC SLAB WAVEGUIDE STRUCTURE… J. NANO- ELECTRON. PHYS. 6, 02007 (2014) 
 
 
02007-3 
served from the figure is the shift of the field profile 
towards the substrate layer as the absolute value of 
Re(εs) or Re(s) increases. Figure 6 shows the same be-
havior as that observed in Fig. 5, but with the cladding 
parameters. The field peak moves towards the film-
cladding interface as the absolute value of Re(εc) or 
Re(c) increases leading to the enhancement of the ev-
anescent field in the cladding layer. This enhancement 
is of great benefit for optical waveguide sensor applica-
tions [17-19]. We now turn our attention the effect of 
imaginary parts of ε and  of the three layers constitut-
ing the waveguide structure under consideration. The 
imaginary parts of ε and  are of great significance in 
LHMs since lossless LHMs are not realistic. The 
electric field distribution in the waveguide is shown 
in Fig. 7 for different values of Im(εf). The mode or-
der (m) in Eq. (7) represents the number of intersec-
tions between the field curve and the x-axis in the 
guiding layer region. When the field curve does not 
intersect the x-axis in the guiding layer, the funda-
mental mode exists. As the figure shows, for 
Im(εf)  0.01, 0.05, and 0.1, the guided mode is the 
fundamental one. There is a transition from the fun-
damental mode (TE0) to the first mode (TE1) as the 
Im(εf) increases to 0.15 and 0.20. This is a significant 
feature showing the importance of the imaginary 
part of εf. As the Im(εf) increases, an additional 
phase shift is imposed on the guided wave. This 
phase condition changes the fundamental mode 
(m  0) to the first mode m  1. Figure 8 shows the 
same behavior when increasing the imaginary part 
of f. The fundamental mode profile in the film layer 
shifts to the next mode when the Im(f) exceeds a 
specific value. 
The electric field distribution of guided modes TE0, 
TE1, TE2, and TE3 are illustrated in Fig. 9. Their prop-
erties are very similar to those of conventional guided 
modes in normal dielectric waveguides. In all these 
modes, the electric field is oscillatory in the guiding 
layer and evanescent in the substrate and cladding 
layers. As mentioned before, the number of intersec-
tions between the field curve and the x-axis in the film 
layer is equal to the mode order. The figure also shows 
that the transverse distribution peaks are asymmetric 
because of asymmetric structure assumed. 
 
0 2 4 6 8 10 12 14
-8
-6
-4
-2
0
2
4
6
8
 F
(h
d
)
 
ta
n
(h
d
)
hd  
 
Fig. 2 – Graphical determination of hd for oscillating guided 
modes of an asymmetric LHM waveguide 
 
 
Finally, we investigate the properties of p-polarized 
light. Figure 10 depicts the transverse field profiles of the 
fundamental modes TE0 and TM0. No remarkable differ-
ences can be found between the two polarizations. The 
field is oscillatory in the film layer while it becomes eva-
nescent in the surrounding media. A little enhancement is 
observed in the field peak for TM0 mode.  
 
-6 -4 -2 0 2 4 6 8 10 12
0.0
0.5
1.0
1.5
2.0
2.5
3.0
 
 
E
y(
x)
x (m)
   = 4.6 GHz
   = 4.8 GHz
   = 5.0 GHz
Substrate Film Cladding
 
 
Fig. 3 – Transverse profiles of the fundamental mode (TE0) for 
different values of the operating frequency for s  f  c  –
 1 + 0.001i, εs  – 4.7 + 0.001i, εf  – 5.7 + 0.001i, and εc  –
 3.7 + 0.001i 
 
-6 -3 0 3 6 9 12
0
1
2
3
4
 
Re
f
5.7
Re
f
6.7
E
y(
x)
x (m)
Substrate CladdingFilm
-6 -3 0 3 6 9 12
0
1
2
3
4
5
 
 
 Re(
f
) = -1
 Re(
f
) = -1.2
x (m)
Substrate Film Cladding
 
 
Fig. 4 – Transverse profiles of the fundamental mode (TE0) for 
different values of the permittivity (left) and permeability 
(right) of the film for s  c  – 1 + 0.001i, εs  – 4.7 + 0.001i 
and εc  – 3.7 + 0.001i 
 
-6 -3 0 3 6 9 12
0
1
2
3
 
 
x (m)
 Re(
s
) = - 4.7
 Re(
s
) = - 5.2
E
y(
x)
x (m)
Substrate Film Cladding
-6 -3 0 3 6 9 12
0
1
2
3
 
 
 
Substrate Film Cladding
 Re(
s
) = - 1
 Re(
s
) = - 1.2
 
 
Fig. 5 – Transverse profiles of the fundamental mode (TE0) for 
different values of the permittivity (Left) and permeability 
(right) of the substrate for f  c  – 1 + 0.001i, εf  –
 5.7 + 0.001i, and εc  – 3.7 + 0.001i 
 
 S.A. TAYA, KH.Y. ELWASIFE J. NANO- ELECTRON. PHYS. 6, 02007 (2014) 
 
 
02007-4 
-6 -3 0 3 6 9 12
0
1
2
3
 
 Re
c
) = - 3.7
 Re
c
) = - 4.5
E
y(
x)
x (m)
Substrate Film Cladding
-6 -3 0 3 6 9 12
0
1
2
3
 
 
Substrate Film Cladding
 Re
c
) = - 1
 Re
c
) = - 1.2
x(m)  
 
Fig. 6 – Transverse profiles of the fundamental mode (TE0) for 
different values of the permittivity (left) and permeability 
(right) of the cladding for s  f – 1 + 0.001i, εf  –
 5.7 + 0.001i, and εs  – 4.7 + 0.001i 
 
-6 -3 0 3 6 9 12
-2
-1
0
1
2
3
E
y(
x)
x (m)
 Im(f) = 0.01
 Im(f) = 0.05
 Im(f) = 0.10
 Im(f) = 0.15
 Im(f) = 0.20
Substrate
Film
Cladding
 
 
Fig. 7 – Transverse profiles of the fundamental mode (TE0) for 
different values of the imaginary part of εf for ω  4.6 GHz, 
s  f  c  – 1 + 0.001i, εs  – 4.7 + 0.001i, and εc  – 3.7 + 0.001i 
 
-6 -3 0 3 6 9 12
-2
-1
0
1
2
CladdingFilm
E
(x
)
x (m)
 Im(f) = 0.01
 Im(f) = 0.02
 Im(f) = 0.03
 Im(f) = 0.04
Substrate
 
 
Fig. 8 – Transverse profiles of the fundamental mode (TE0) for 
different values of the imaginary part of f for ω  4.6 GHz, 
s  c  – 1 + 0.001i, εs  – 4.7 + 0.001i, εf  – 5.7 + 0.001i and 
εc  – 3.7 + 0.001i 
-10 -5 0 5 10 15 20
-2
-1
0
1
2
3
4
5
Film
Cladding  
 
E
y(
x)
x (m)
 TE0
 TE1
 TE2
 TE3
Substrate
 
 
Fig. 9 – Transverse profiles of different mode (TE) orders for 
ω  4.6 GHz, s  f  c  – 1 + 0.001i, εs  – 4.7 + 0.001i, εf  –
 5.7 + 0.001i, and εs  – 4.7 + 0.001i  
 
-6 -4 -2 0 2 4 6 8 10 12
0.0
0.5
1.0
1.5
2.0
2.5
3.0
CladdingSubstrate Film
 
 
E
y(
x)
x (m)
 TE
TM 
 
 
Fig. 10 – Transverse profiles of TE0 and TM0 polarizations for 
ω  4.6 GHz, s  f  c  – 1 + 0.001i, εs  – 4.7 + 0.001i, εf  –
 5.7 + 0.001i, and εc  – 3.7 + 0.001i  
 
4. CONCLUSION 
 
Asymmetric three-layer slab waveguide structure was 
considered. All layers were assumed to have a negative 
index of refraction. The propagation of electromagnetic 
waves in such a structure was treated. It was found 
that the structure can support guided modes of any 
order. The fundamental mode can exist as well as high-
er modes. Moreover, it was found that increasing the 
imaginary part of the guiding layer permittivity or 
permeability change the fundamental mode to the first 
mode. 
 
 
 
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(2006). 
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English

Field Profile of Asymmetric Slab Waveguide Structure with LHM Layers

Electronic Sumy State University Institutional Repository

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ Vol. 6 No 2, 02007(5pp) (2014) Том 6 № 2, 02007(5cc) (2014)   2077-6772/2014/6(2)02007(5) 02007-1  2014 Sumy State University Field Profile of Asymmetric Slab Waveguide Structure with LHM Layers  S.A. Taya*, Kh.Y. Elwasife†   Physics Department, Islamic University of Gaza, Palestinian Authority  (Received 16 March 2014; published online 20 June 2014)  This paper presents asymmetric three-layer slab waveguide structure in which all layers are consid-ered left-handed materials (LHMs) with negative electric permittivity and magnetic permeability. The electric field profile is investigated in details with all parameters of the waveguide structure. Many inter-esting features are observed such as the existence of the fundamental mode and the transition from the fundamental mode to the first guided mode with increasing the imaginary part of the guiding layer permit-tivity or permeability.  Keywords: Slab waveguides, Guided modes, Left-handed materials.    PACS numbers: 42.65.Wi, 42.65.Tg, 42.25.Bs                                                                   * staya@iugaza.edu.ps † kelwasife@iugaza.edu.ps 1. INTRODUCTION  Several groups have succeeded in fabricating left-handed media and it is now possible to employ left-handed electromagnetism in a large range. In recent years, the topic of metamaterials, i.e. artificial materials synthesized by embedding specific inclusions in host media, has in-creasingly received a renewed attention due to the inter-est in man-made complex materials that may possess negative permittivity (ε) and permeability (μ) in a certain frequency band. In 1968, Veselago postulated theoretically a material in which both ε and μ were assumed to have negative real values and he analyzed plane wave propaga-tion in such a medium [1]. These materials were called left-handed metamaterials (LHMs). Since then an inten-sive research has been carried out to understand the properties of electromagnetic waves in these media [2-20]. I. V. Shadrivov analyzed nonlinear guided waves in a pla-nar waveguide made of a LHM surrounded by a Kerr-like nonlinear dielectric and predicted that such a waveguide can support fast and slow symmetric and antisymmetric nonlinear modes [2]. Moreover, he studied the symmetry breaking bifurcation and asymmetric modes in such a symmetric structure and showed that the modes can be either forward or backward [2]. Characteristics of surface wave modes on a grounded slab of LHM were investigated [3]. It was shown that, unlike a slab with positive parame-ters, the dominant mode can have evanescent fields on both sides of the interface between the slab and the sur-rounding air. The guided and surface modes in an LHM asymmetric slab waveguide were studied by introducing three normalized parameters [4-8]. The guided dispersion characteristics of the fundamental symmetric and asym-metric modes of surface waves along single- and double-negative indexed slab waveguides were investigated and a comparative analysis was made when varying the single- and double-negative permittivity and permeability [9]. Asymmetric waveguide structure with a LHM guiding layer surrounded by a linear dielectric substrate and a gyromagnetic ferrite cladding has been investigated [10]. Guided modes supported by an asymmetric three-layer metal-clad waveguide structure were presented when the guiding layer is LHM [11]. Multilayer planar waveguide structures comprising LHMs have been explored for both TM and TE polarizations [12-14]. TE surface modes guid-ed by a single interface and a slab waveguide containing LHMs were analyzed by using a ray picture [15]. The transfer function of the discretized perfect lens in finite-difference time-domain and transfer matrix method simu-lations was investigated [16]. Waveguide structures com-prising LHMs have been strong candidates for many opto-electronics applications. One of the most interesting appli-cations of LHM waveguide structures is optical sensing [17-20] in which any change in the refractive index of the analyte is detected.  In this paper asymmetric three-layer planar wave-guide structure is studied for both TE and TM modes. All layers are considered to be LHMs. The field profile is investigated in details with all parameters of the LHMs.   2. THEORY  We suppose an asymmetric three-layer slab waveguide structure. The substrate, guiding film, and cladding are assumed to be LHMs with εi  0 and I  0 (i  1, 2, 3). The waves are considered to propagate along z-direction and x-axis is perpendicular to the plane separating the structure layers as shown in Fig. 1.  dzxLHM film ε1, µ1LHM cald ε2, µ2LHM substrate ε3, µ3  Fig. 1 – Cross-sectional view of asymmetric LHM slab waveguide   S.A. TAYA, KH.Y. ELWASIFE J. NANO- ELECTRON. PHYS. 6, 02007 (2014)   02007-2 Since the waveguide is homogeneous along z -axis, solutions to the wave equation can have the form    ˆ( , , ) ( )exp( ( ))y yE x z t E x j t z a   , (1)   ˆ ˆ( , , ) [ ( ) ( ) ]exp( ( ))x x z zH x z t H x a H x a j t z    , (2)  where  is the angular frequency and β is the propaga-tion constant. The wave vector components h, q2, and q3 are defined as 2 20 1 1h k     , 2 22 0 2 2q k    , 2 23 0 3 3q k    . The y-component of the electric field for TE polari-zation is given by    22( sin( ) cos( )) 0< <( ) exp( ) >dexp( ) <0yA hx B hx x dE x C q x xD q x x  (3)  The magnetic field is given by   EHj, (4)  resulting in    ( )yzEjH xx (5)  The boundary conditions require that the tangential components of both E  and H  are continuous across the boundaries. The boundary condition equation for the asymmetric modes is given by   1 3 1 23 21 3 1 23 2tan( )1q qh hhdq qh h     (6)  To introduce the mode order (m), Eq. (6) can be written as   1 11 3 1 23 2tan tanq qhd mh h             . (7)  The right –hand side (RHS) of Eq. (6) can be written as a function of (hd). We first assume 2q and 3q to have the forms   2 22 0 1 1 2 2( )q k h      , (8)   2 23 0 1 1 3 3( )q k h       (9)  with these notation for 1q  and 3q , the RHS of Eq. (6) can be written as    2 2 2 2321 12 2 2 2 22 321( ) ( )( )( ) ( ) ( )s hd p hdF hd hdhd s hd p hd      (10)  where  0 1 1 2 2s k d       and  0 1 1 3 3p k d      . When propagation constant  exceeds a critical val-ue, the wave number h becomes purely imaginary. For TM polarization, the boundary condition equa-tion is given by   1 3 1 23 21 3 1 23 2tan( )1q qh hhdq qh h     (11)  3. RESULTS AND DISCUSSION  Figure 1 shows the slab waveguide geometry under consideration. In the analysis below, the frequency of electromagnetic waves is assumed to be 4.6GHz. Oscil-lating mode can exist in the proposed structure for 2 2 20 1 1 0 3 3> >k k     . Graphical method can be used to determine the values of hd for possible guided modes. Tan (hd) given by Eq. (6) and F(hd) given by Eq.(10) are plotted versus hd in Fig. 2. The intersection points give the values of hd at which the guided modes exist. It is clear from the figure that the proposed structure can support guided modes. Transverse profiles of the TE0 mode for different values of the operating frequen-cy are plotted in Fig. 3 for s  f  c  – 1 + 0.001i, εs  – 4.7 + 0.001i, εf  – 5.7 + 0.001i, and εc   – 3.7 + 0.001i. The guiding layer thickness is assumed to be 6 m. As can be seen from the figure, the electric field oscillates in the guiding layer and is evanescent in the surrounding media. The two tails of the evanescent are asymmetric in the substrate and cladding due to the asymmetric structure assumed. Increasing the an-gular frequency of the guided wave from 4.6 GHz to 5 GHz does not have a considerable effect on the field profile. It merely enhances the peak of the oscillating waves in the guiding layer. Moreover, it causes a barely detectable reduction in the evanescent field in the sur-rounding media. It is worth to mention that the wave-guide structure considered in this work can accommo-date the fundamental mode. The absence of the funda-mental mode was reported in many works in literature [7] when a LHM guiding layer is surrounded by two linear dielectric media. Our investigation here of a LHM guiding layer surrounded by two LHMs empha-sizes that this structure can support any mode includ-ing the fundamental one. Transverse profiles of the fundamental mode for different values of the permittiv-ity (left) and permeability (right) of the film are shown in Fig. 4. As can be seen from the figure, increasing the absolute value of Re(εf) or Re(f) reduces the evanescent field in the substrate and cladding layers. Moreover, it leads to an enhancement of the oscillating wave peak in the guiding layer. This property is very similar to that of conventional guided modes in the well-known dielec-tric waveguides in which the increasing of the guiding layer refractive index enhances the confinement of the wave and reduces the wave tails in the surrounding media. The distributions of electric field amplitude in the waveguide for different values of the substrate permittivity (left) and permeability (right) are shown in Fig. 5. The most important feature that can be ob- FIELD PROFILE OF ASYMMETRIC SLAB WAVEGUIDE STRUCTURE… J. NANO- ELECTRON. PHYS. 6, 02007 (2014)   02007-3 served from the figure is the shift of the field profile towards the substrate layer as the absolute value of Re(εs) or Re(s) increases. Figure 6 shows the same be-havior as that observed in Fig. 5, but with the cladding parameters. The field peak moves towards the film-cladding interface as the absolute value of Re(εc) or Re(c) increases leading to the enhancement of the ev-anescent field in the cladding layer. This enhancement is of great benefit for optical waveguide sensor applica-tions [17-19]. We now turn our attention the effect of imaginary parts of ε and  of the three layers constitut-ing the waveguide structure under consideration. The imaginary parts of ε and  are of great significance in LHMs since lossless LHMs are not realistic. The electric field distribution in the waveguide is shown in Fig. 7 for different values of Im(εf). The mode or-der (m) in Eq. (7) represents the number of intersec-tions between the field curve and the x-axis in the guiding layer region. When the field curve does not intersect the x-axis in the guiding layer, the funda-mental mode exists. As the figure shows, for Im(εf)  0.01, 0.05, and 0.1, the guided mode is the fundamental one. There is a transition from the fun-damental mode (TE0) to the first mode (TE1) as the Im(εf) increases to 0.15 and 0.20. This is a significant feature showing the importance of the imaginary part of εf. As the Im(εf) increases, an additional phase shift is imposed on the guided wave. This phase condition changes the fundamental mode (m  0) to the first mode m  1. Figure 8 shows the same behavior when increasing the imaginary part of f. The fundamental mode profile in the film layer shifts to the next mode when the Im(f) exceeds a specific value. The electric field distribution of guided modes TE0, TE1, TE2, and TE3 are illustrated in Fig. 9. Their prop-erties are very similar to those of conventional guided modes in normal dielectric waveguides. In all these modes, the electric field is oscillatory in the guiding layer and evanescent in the substrate and cladding layers. As mentioned before, the number of intersec-tions between the field curve and the x-axis in the film layer is equal to the mode order. The figure also shows that the transverse distribution peaks are asymmetric because of asymmetric structure assumed.  0 2 4 6 8 10 12 14-8-6-4-202468 F(hd) tan(hd)hd   Fig. 2 – Graphical determination of hd for oscillating guided modes of an asymmetric LHM waveguide   Finally, we investigate the properties of p-polarized light. Figure 10 depicts the transverse field profiles of the fundamental modes TE0 and TM0. No remarkable differ-ences can be found between the two polarizations. The field is oscillatory in the film layer while it becomes eva-nescent in the surrounding media. A little enhancement is observed in the field peak for TM0 mode.   -6 -4 -2 0 2 4 6 8 10 120.00.51.01.52.02.53.0  Ey(x)x (m)   = 4.6 GHz   = 4.8 GHz   = 5.0 GHzSubstrate Film Cladding  Fig. 3 – Transverse profiles of the fundamental mode (TE0) for different values of the operating frequency for s  f  c  – 1 + 0.001i, εs  – 4.7 + 0.001i, εf  – 5.7 + 0.001i, and εc  – 3.7 + 0.001i  -6 -3 0 3 6 9 1201234 Ref5.7Ref6.7Ey(x)x (m)Substrate CladdingFilm-6 -3 0 3 6 9 12012345   Re(f) = -1 Re(f) = -1.2x (m)Substrate Film Cladding  Fig. 4 – Transverse profiles of the fundamental mode (TE0) for different values of the permittivity (left) and permeability (right) of the film for s  c  – 1 + 0.001i, εs  – 4.7 + 0.001i and εc  – 3.7 + 0.001i  -6 -3 0 3 6 9 120123  x (m) Re(s) = - 4.7 Re(s) = - 5.2Ey(x)x (m)Substrate Film Cladding-6 -3 0 3 6 9 120123   Substrate Film Cladding Re(s) = - 1 Re(s) = - 1.2  Fig. 5 – Transverse profiles of the fundamental mode (TE0) for different values of the permittivity (Left) and permeability (right) of the substrate for f  c  – 1 + 0.001i, εf  – 5.7 + 0.001i, and εc  – 3.7 + 0.001i   S.A. TAYA, KH.Y. ELWASIFE J. NANO- ELECTRON. PHYS. 6, 02007 (2014)   02007-4 -6 -3 0 3 6 9 120123  Rec) = - 3.7 Rec) = - 4.5Ey(x)x (m)Substrate Film Cladding-6 -3 0 3 6 9 120123  Substrate Film Cladding Rec) = - 1 Rec) = - 1.2x(m)   Fig. 6 – Transverse profiles of the fundamental mode (TE0) for different values of the permittivity (left) and permeability (right) of the cladding for s  f – 1 + 0.001i, εf  – 5.7 + 0.001i, and εs  – 4.7 + 0.001i  -6 -3 0 3 6 9 12-2-10123Ey(x)x (m) Im(f) = 0.01 Im(f) = 0.05 Im(f) = 0.10 Im(f) = 0.15 Im(f) = 0.20SubstrateFilmCladding  Fig. 7 – Transverse profiles of the fundamental mode (TE0) for different values of the imaginary part of εf for ω  4.6 GHz, s  f  c  – 1 + 0.001i, εs  – 4.7 + 0.001i, and εc  – 3.7 + 0.001i  -6 -3 0 3 6 9 12-2-1012CladdingFilmE(x)x (m) Im(f) = 0.01 Im(f) = 0.02 Im(f) = 0.03 Im(f) = 0.04Substrate  Fig. 8 – Transverse profiles of the fundamental mode (TE0) for different values of the imaginary part of f for ω  4.6 GHz, s  c  – 1 + 0.001i, εs  – 4.7 + 0.001i, εf  – 5.7 + 0.001i and εc  – 3.7 + 0.001i -10 -5 0 5 10 15 20-2-1012345FilmCladding   Ey(x)x (m) TE0 TE1 TE2 TE3Substrate  Fig. 9 – Transverse profiles of different mode (TE) orders for ω  4.6 GHz, s  f  c  – 1 + 0.001i, εs  – 4.7 + 0.001i, εf  – 5.7 + 0.001i, and εs  – 4.7 + 0.001i   -6 -4 -2 0 2 4 6 8 10 120.00.51.01.52.02.53.0CladdingSubstrate Film  Ey(x)x (m) TETM   Fig. 10 – Transverse profiles of TE0 and TM0 polarizations for ω  4.6 GHz, s  f  c  – 1 + 0.001i, εs  – 4.7 + 0.001i, εf  – 5.7 + 0.001i, and εc  – 3.7 + 0.001i   4. CONCLUSION  Asymmetric three-layer slab waveguide structure was considered. All layers were assumed to have a negative index of refraction. The propagation of electromagnetic waves in such a structure was treated. It was found that the structure can support guided modes of any order. The fundamental mode can exist as well as high-er modes. Moreover, it was found that increasing the imaginary part of the guiding layer permittivity or permeability change the fundamental mode to the first mode.    REFERENCES  1. V.G. Veselago, Sov. Phys. 10 No 4, 509 (1968). 2. Ilya V. Shadrivov, Photonics and Nanostructures – Fun-damentals and Applications 2,175 ( 2004). 3. S. Mahmoud, A.J. Viitanen, Prog. Electromagnetic Res. 51, 127 (2005). 4. Z.J. Wang, J.F. Dong, Prog. Electromagnetics Res. 62, 203 (2006). 5. S.A. Taya, I.M. Qadoura, Optik- Int. J. Light Electron Opt. 124 No 13, 1431 (2012). 6. I.M. Qadoura, S.A. Taya, K.Y. El-Wasife IJMOT 7 No 5, 349 (2012). 7. S.A. Taya, H.M. Kullab, I.M. 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Field Profile of Asymmetric Slab Waveguide Structure with LHM Layers

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