General Formulation to Investigate Scattering from
Multilayer Lossy Inhomogeneous Metamaterial Planar
Structures
Mohammad Kiani*, Ali Abdolali**
Abstract: This paper presents a general formulation to investigate the scattering from
Multilayer Lossy Inhomogeneous Metamaterial Planar Structure (MLIMPS) with arbitrary
number of layers and polarization. First, the dominating differential equation of transverse
components of electromagnetic fields in each layers derived. Considering the general form
of solution of the differential equations and the boundary conditions of the problem; a set of
linear equations is obtained. By solving these equations, the electromagnetic fields in all
layers and reflection and transmission coefficients are calculated. This method is applied in
an interesting example for two bi-layered structures with inhomogeneous conventional
material and metamaterial profile for constitutive parameters. Results which are presented
in example are useful for constructing general duality between conventional material and
metamaterials.
Keywords Multilayer Lossy Inhomogeneous Metamaterial Planar Structure (MLIMPS),
general formulation, appealing phenomena.
1 Introduction 1
Homogeneous planar layers are the most applicable
structures in filters, shielding and etc. Inhomogeneous
materials in these structures can optimize shielding
effectiveness and provide less scattering and larger or
smaller bandwidth [1-7]. There are different methods to
analysis inhomogeneous planar multilayer structure
such as Taylor’s series expansion [8], Fourier’s series
expansion [9], Equivalent source method [10], Method
of moment [11], Inverse scattering method [12] and so
on. All this methods are numerical and suggested for a
slab of inhomogeneous medium. In this paper, a general
formulation for investigating scattering from arbitrary
Multilayer Lossy Inhomogeneous Planar Structure
(MLIMPS) with arbitrary number of layers and
polarization is presented. Dominating differential
equations of MLIMPS are written in form of linear
second order differential equation with non-constant
coefficient. General form of solution of these equations
is presented. Using boundary conditions of the problem,
a set of linear equation is obtained which leads to
* Mohammad Kiani is the student of M.Sc. in Department of
Electrical Engineering, Iran University of Science and Technology,
Tehran-Narmak
E-mail:Mohammadkiani@elec.iust.ac.ir (co-author)
** Ali Abdolali is assistant professor in in Department of Electrical
Engineering, Iran University of Science and Technology, TehranNarmak
E-mail:Abdolali@iust.ac.ir (corresponding Author)
calculating the electromagnetic fields in all layers and
reflection and transmission coefficients. This
formulation is applicable in stating general
fundamentals
and
properties
of
multilayer
inhomogeneous metamaterial structures and can be used
for providing conditions about appealing phenomena for
mentioned structures as zero reflection, zero
transmission, cloaking and etc. Finally, presented
method is applied in interesting example for two
conventional and metamaterial structures which are
useful for research about duality between conventional
materials and metamaterials in general case [13].
2 Differential Equation of MLIMPS
Fig.1 indicates atypical MLIMPS with N layers. The
left medium is free space and the right is arbitrary
homogeneous medium. Constitutive parameters of kth
layer as follow
(1)
 ( z)   ( z)  j ( z)
k
Rk
Ik
 k ( z )   Rk ( z )  j Ik ( z )
(2)
k  1, 2,3,...., N
By combination of Maxwell’s Curl equations, we can
write vector wave equation for kth as follow [14]
1
2
(3)
  ( ( z)    E )    ( z)E
k
k
k
k
  ( k ( z )    H k )    k ( z ) H k
1
2
(4)
Because of the infinite dimension of structure in y
direction, we have

 jK x x

or
x
(5)
0
Q ( x, z )  P ( z )e
(6)
  jK x
Where
y
Under phase-matching condition in the boundary of
each two consecutive medium, we have
(7)
K x  K 0 sin  i
K0  
(8)
 0 0
Fig.1 lossy inhomogeneous planar multilayer structure a) TEz b) TMz
 Eky ( x, z )  Eky ( z )e  jK x

 jK x
 H ky ( x, z )  H ky ( z )e
TE
x
In above equation, indicates the general form of
electromagnetic field in each medium
Using (3) and (4) and notice to (5) and (6), differential
equation of transverse component of electromagnetic
fields in kth medium can be obtained as below
z
For TE polarization
2
d E ky ( z )
dz
2

 'k ( z ) dE ky ( z )
k ( z )

dz
(9)
Equations (9) and (11) are linear second order
differential equation with non-constant coefficients.
From principles of ordinary differential equations,
second order linear differential equation of transverse
fields in kth inhomogeneous medium has general
solution as follows
Ck ,1 f k ( z )  Ck ,2 g k ( z ) 
2
(  k ( z ) k ( z )  K x ) E ky ( z )  0
2
 'k ( z ) 
(10)
dz
2

dz
 k '( z ) dH ky ( z )
 k (z)
dz
(11)
2
 (  k ( z ) k ( z )  K x ) H ky ( z )  0
Where
2
 k '( z ) 
Where
d k ( z)
dz
TM
(14)
(12)
Ck ,1 and Ck ,2
electromagnetic fields in kth medium and
z
2
TE
In (14), f k and g k show the variation of transverse
For TM polarization
d H ky ( z )
 Eky ( z )

 H ky ( z )
k  1, 2, ......, N
Where
d k ( z )
(13)
TM
x
are unknown constant coefficients which can be
calculated by using the boundary conditions of
tangential component of fields at interface of kth and
k+1th medium.
2.2
Boundary Conditions
Considering the general form of transverse fields in
equation (14), the boundary conditions for tangential
component of electric and magnetic fields in each
consecutive layer could be written as follow
at boundary of free-space and first inhomogeneous
medium ( z  0 )
i
F t ( z ) z 0  F
r
t
( z ) z 0

( C1,1 f1 ( z )
 C
1,2
g ( z ))
1
z 0
(15)
F
i
t
(z )  F
r
t
( z ) z 0

i

 E y ( z ) TE
F t ( z)  
(24)
r

 E y ( z ) TE
F t ( z)  
r

 H y ( z ) TM
(25)
i
(16)
i

 H y ( z ) TM
IS
j
'
 p1 ( z )
'
(C f 1 ( z )  C g 1 ( z )) z  0
1 ,1
r
1,2
at boundary of kth and k+1th layers (
z  d k , k , k  1, 2,...., N  1 )
T
(C k ,1 f k ( z )  C k ,2 g k ( z )) z d 
C k 1,1 f k 1 ( z )  C k 1,2 g k 1 ( z ) z d
[
pk ( z )
(27)
F t ( x, z )  F t ( z )e
 jK x x
(28)
F t ( x, z )  F t ( z )e
 jK x x
i
'
(C k ,1 fk ( z )  C k ,2 g k ( z ))] z  d
i
r
k
T
(18)
1
[
 jK x x
F t ( x, z )  F t ( z )e
k
r
'
'
pk ( z )
T
2.3
Matrix Form for Equations
With introducing reflection and transmission coefficient
as follows
k
r

N
N
(19)
T
 pN ( z)
'
'
( C f ( z )  C g ( z ))] z  d
N ,1
N
N ,2
N

F t ( z ) z d
N
IL
F t (0)
(30)
i
F t (0)
T
C N ,1 f N ( z )  C N ,2 g N ( z ) z d  F t ( z ) z d
[
(29)
'
C k 1,1 fk 1 ( z )  C k 1,2 g k ( z )] z  d
and at boundary of Nth inhomogeneous layer and the
most right half space medium( z  d N )
j
(26)
T

 H y ( z ) TM
(17)
k
1
T

 E y ( z ) TE
F t ( z)  
N
(20)

F T t (d N )
(31)
F i t (0)
We can rewrite equations (15)-(20) as follow
At boundary of free-space and first inhomogeneous
medium ( z  0 )
i
F t (0)(1   )  ( C1,1 f1 ( z )  C 2 ,1 g1 ( z )) z  0
Where
 0
 cos  TE
i
IS  
 1
TM
 0 cos  i
 1
 cos  TE
t
IL  
 1
TM
1 cos  t
  k ( z ) TE
pk ( z )  
 k ( z )
TM
i
F t (0)(1   )
I
(21)
S

j
'
 p1 ( z )
(32)
'
(C1,1 f1 ( z )  C 2 ,1 g1 ( z )) z 0
(33)
At boundary of kth and k+1th layers (
z  d k , k , k  1, 2,...., N  1 )
(Ck ,1 fk ( z )  Ck ,2 g k ( z )) z  d  C k 1,1 fk 1 ( z )  C k 1,2 g k 1 ( z ) z d
k
(22)
1
[
'
pk ( z )
(23)
[
'
(C k ,1 fk ( z )  C k ,2 g k ( z ))] z  d
1
pk ( z )
k
(35)
'
'
C k 1,1 fk 1 ( z )  C k 1,2 g k ( z )] z  d
k
k
(34)
At boundary of Nth inhomogeneous and right medium (
z  dN )
Ck ,1 , Ck ,2
1 k  N
i
C f ( z )  C g ( z ) z  d  F t ( z ) z  d
N ,1
N
N ,2
N
N
F t ( z ) z  d
i
j
[
 pN ( z)
'
'
( C f ( z )  C g ( z ))] z  d 
N ,1
N
N ,2
N
N
I
, 
(36)
N
N
(37)
L
Using equations (32)-(37) leads to 2N+2 linear
equations with 2N+2 unknown variables which can be
solve easily to obtain reflection and transmission
coefficients and electromagnetic fields in all regions.
These unknown variables can calculate as follow
Therefor a set of linear equations can be written as a
suitable matrix of equation as follows
This matrix form of equations is applicable and so
useful for analysis of the arbitrary type of MLIMPS and
this method can introduce as full-wave method for
analysis of MILPS.As a potential application, this
method can be led researchers to interesting phenomena
as zero reflection and zero transmission and can be
construct the novel duality between conventional
material and metamaterials in general case.
  F (0) f1 ( z ) z 0 g1 ( z ) z 0 0
0
0
 i
'
'
jg1 ( z )
 F (0) jf1 ( z )
0
0
0
 I S  p1 ( z ) z 0  p1 ( z ) z 0

 0

0
0
f k ( z ) z dk g k ( z ) z dk  f k 1 ( z ) z dk  g k 1 ( z ) z dk
 0

fk ' ( z)
gk ' ( z)
 f k 1' ( z )
 g k 1' ( z )
 0
0
0
z d
z d
z d
z d

pk ( z ) k pk ( z ) k pk 1 ( z ) k pk 1 ( z ) k

 0
 0
0
0


 0
0
0
0
0

i
(38)
0
0
0
0
0
0
0
0
0
0
0
0
0
f N ( z ) z dN
g N ( z ) z d N
0
jf N ' ( z )
 pN ( z )
z dN
jg N ' ( z )
 pN ( z )
z dN
 F i (0) 
0     i 
  C   F (0) 
1,1 
 I 
0  
 C1,2   S 
  0 

 


  Ck ,1  

0 
  0 
  Ck ,2   
0 
0   Ck 1,1  
  0 

 Ck 1,2  

  0 




 F i (0)   C  
 N ,1 

0 


 F i (0)   CN ,2  

 0 
I L     

 0 
(39)
3 Example and Discussion
In this section, two inhomogeneous planar bi-layered
structures with conventional material and metamaterial
are considered to analyze as follow. Media of two halfspaces in both structures are free space.
Right
inhomogeneous
layer
Left
inhomogeneous
layer
Structure 1
r  1
Structure 2
 r  1
r  z  2
r  z  2
r  1
 r  1
 r  3z  1
 r  3 z  1
The exact solution of Eq. (9) for above constitutive
parameters is presented in Appendix A. Structures are
illuminated with TEz polarization incident plane wave,
incidence angle 60, excitation frequency 1GHz, and
amplitude of electric field E  1V / m . Thickness of
each layer is 20cm.Fig. 2 and Fig. 3 represent the
amplitude of transverse component of electric field in
both structures in first and second inhomogeneous layer
respectively.
i
clearly see that the reflected and transmitted powers for
1
 jK x
Structure 1
Structure 2
0.9
both structures are same. Since the term e x is
common for all fields components, then, the electric and
magnetic fields in two structures are not complex
conjugate of each other which is an appealing result it
can lead to research about duality between conventional
and metamaterial inhomogeneous structures in general
case [15].
0.8
|Ey1| (V/m)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
z (m)
1
Fig.2 Amplitude of transverse electric field in the first layer for both
structures at f  1.0 GHz
0.9
0.8
0.7
| |
0.6
0.16
Structure 1
Structure 2
0.14
0.5
0.4
0.3
Structure 1
Structure 2
0.12
|Ey2| (V/m)
0.2
0.1
0.1
0.08
0
0
10
20
30
0.06
40
50
 i (Degree)
60
70
80
90
Fig.6The Amplitude of reflection coefficient for various incident
angle for both structures at f  1.0 GHz
0.04
0.02
0
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
0.36
0.38
0.4
z (m)
1
Fig.3The amplitude of transverse electric field in second layer for second and
first structure
0.9
0.8
Structure 1
Structure 2
0.7
0.6
|T|
Fig. 4 and Fig. 5 show amplitude of magnetic field
transverse component for two structures in first and
second inhomogeneous layer respectively.
0.5
0.4
0.3
-3
3
x 10
0.2
Structure 1
Structure 2
2.5
0.1
0
|Hx1| (A/m)
2
0
10
20
30
40
50
 i (Degree)
60
70
80
90
Fig.7The amplitude of transmission coefficient for various incident
angle for second and first structure at f  1.0 GHz
1.5
1
0.5
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
z (m)
Fig.4Theamplitude of transverse magnetic field in first layer for second and first
structure at f  1.0 GHz
-4
x 10
Structure 1
Structure 2
|Hx2| (A/m)
3
2
1
0
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
0.36
0.38
0.4
z (m)
Fig.5 Amplitude of transverse magnetic field in first layer for second and first
structure at f  1.0 GHz
Fig. 6 and Fig. 7 represent amplitude of reflection and
transmission for two structures respectively. It can
5 Conclusion
General formulation to investigate the scattering from
Multilayer Lossy Inhomogeneous Metamaterial Planar
Structure (MLIMPS) was presented. With using
Maxwell’s equations and notice to physical and
geometrical properties of structure and plane wave
which illuminate the assumed structure, dominating
differential equations of MLIMPS were derived. From
principles of ordinary differential equations, the general
form of solution of wave equation for transverse
component of electric and magnetic field is presented
and with this general solution and using the boundary
conditions of problem, a set of linear equations was
obtained. Solving these equations, reflection and
transmission coefficient and electromagnetic fields in all
regions were calculated. Also this method which
introduced is applicable for all frequencies especially
for very low or high frequencies. Finally proposed
method has been used as an interesting example for two
conventional material structure and metamaterial
structure which can be useful for research about the
constructing general duality between conventional
materials and metamaterials.
Appendix A
In this section, the exact electric and magnetic fields in
first and second structure are obtained. By using
following electric and magnetic parameters in Eq. 9, one
can obtain the following differential equation for E y
 r  a  bz
A1
r  1
A2
[4]
[5]
[6]
2
d
dz
2
E y  ( A  Bz ) E y  0
A3
[7]
Where
A    0 0a  K x
A4
B    0 0b
A5
2
2
2
The above equation can be simplified with change of
variable by letting
A
1/ 3
  B (z  )
A6
B
By using this change, Eq. A3 will become
[8]
[9]
2
d Ey
  E y  0 A7
2
d
The above equation is known as the Airy equation, and
the corresponding solution are Airy function A i (  )
and B i ( ) , Hence
[10]
[11]
E y (z )  C 1 A i (  )  C 2 B i (  ) A
8
Where A i (  ) and B i ( ) are Airy functions of first
[12]
and second type.
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