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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES
1
Determination of Effective Constitutive Parameters
of Inhomogeneous Metamaterials With Bianisotropy
Ugur Cem Hasar , Member, IEEE, Gul Buldu , Yunus Kaya, and Gokhan Ozturk
Abstract— A recursive algorithm is proposed to accurately
extract the electromagnetic parameters of isotropic/bianisotropic
metamaterial (MM) slabs of inhomogeneous structures.
The algorithm is based on the recursive calculations of measured/
simulated forward and backward scattering (S-) parameters from
previously extracted wave impedances and refractive indices
of each homogeneous layer (except the analyzed intermediate
layer) by a suitable retrieval method. There are two main
advantages of our proposed method over existing extraction
algorithms. First, it could retrieve the electromagnetic parameters
of both isotropic and bianisotropic MM slabs. Second, it is
applicable for electromagnetic parameter extraction of inhomogeneous structures. For validation of our method and assessing its
accuracy over existing extraction methods, we used the simulation
results of various inhomogeneous structures constructed by
different isotropic/bianisotropic MM slabs (split-ring-resonator
(SRR), SRR-wire, omega, and omega-wire). For completeness,
the effects of noise present in S-parameters and iterative error
(by increasing the value of iteration numbers) are examined to
test the robustness of the proposed algorithm.
Index Terms— Bianisotropic, constitutive parameters, inhomogeneous, metamaterials (MMs), recursive algorithm, retrieval.
I. I NTRODUCTION
T
HE electromagnetic response of materials is generally
identified by the parameters of relative permittivity (εr )
and relative permeability (μr ) [or similar terms such as magnetoelectric coupling coefficient (ξ0 ) and chirality parameter (κ)]
or wave impedance (Z ) and refractive index (n). In order to
interpret the electromagnetic response of conventional materials (right-handed materials) or engineered materials [lefthanded materials generally coined as metamaterials (MMs)]
and apply these materials for some specific applications
(e.g., electromagnetic cloak [1] and perfect lens [2]), their
electromagnetic parameters should be accurately retrieved.
Manuscript received September 25, 2017; revised January 19, 2018;
accepted March 6, 2018. This work was supported by the Scientific and Technological Research Council of Turkey under Project 112R032. (Corresponding
author: Ugur Cem Hasar.)
U. C. Hasar is with the Department of Electrical and Electronics
Engineering, Gaziantep University, 27310 Gaziantep, Turkey (e-mail:
uchasar@gantep.edu.tr).
G. Buldu is with the Department of Electrical and Electronics Engineering,
Munzur University, 62000 Tunceli, Turkey.
Y. Kaya is with the Department of Electricity and Energy, Bayburt
University, 69000 Bayburt, Turkey.
G. Ozturk is with the Department of Electrical and Electronics Engineering,
Ataturk University, 25240 Erzurum, Turkey.
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMTT.2018.2846726
In this paper, our focus is to accurately extract electromagnetic
properties of MM structures.
Generally, homogenization approach is applied to MMs [3],
when the operating wavelength is much larger (long wavelength limit) than the lattice periods of MMs (quasi-static
case), to make the complex electromagnetic response of MMs
to be equivalent to the response of a hypothetical continuous material. Various equivalence models were defined such
as external equivalence, dispersion equivalence, and singlemode or double-mode equivalences, considering the completeness and sophistication of the constructed model [4].
Among these models, our primary concern is to utilize the
external equivalence approach through which effective electromagnetic properties of MM slabs can be retrieved from
scattering (S-) parameters. There are already many techniques
available in the literature for extraction of electromagnetic
properties of isotropic, bianisotropic, and chiral MM slabs
based on measured/simulated free space (normal or oblique
incidence) or waveguide S-parameters with/without boundary
effects [5]–[38]. All these studies were restricted to retrieval
of electromagnetic properties of homogeneous MM slabs. The
methods in [39] and [40] can be applied for electromagnetic
parameter retrieval of inhomogeneous MM slabs. However,
these methods do not consider the effect of magnetoelectric
coupling arising from the coupling between electric and magnetic fields and thus not applicable for retrieval of electromagnetic properties of bianisotropic MM slabs. The unique
advantages of this type of MM slabs are that their n could have
a positive real part for a backward wave propagation [41], they
have a wider stopband [42], and they have different forward
and backward wave impedances resulting in different forward and backward S-parameters [17], different forward and
backward powers [43], and different semi-infinite reflection
coefficients [30]. In this paper, we extend the methodologies
in [39] and [40] and propose a retrieval procedure for the determination of electromagnetic properties of bianisotropic inhomogeneous MM slabs using measured/simulated S-parameters.
To this end, we will consider the whole inhomogeneous
bianisotropic MM slab as a cascade connection of piecewise
bianisotropic homogeneous MM slabs [44].
The organization of the rest of this paper is as follows.
First, we give a theoretical analysis to derive intermediate
recursive S-parameters in terms of forward and backward wave
impedances and refractive indices of each homogeneous layer
(e.g., bianisotropic MM slab) of the whole inhomogeneous
structure in Section II. Then, in Section III, we present our
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES
where dt = 0 for s = 0 and for s ≥ 1
dt =
s
= dκ .
(3)
κ=1
Besides, for the m + 1 layer, electric and magnetic fields are
i
Ē m+1 = āx E m+1
e+ j k0 nm+1 (z−(d1+d2 +···+dm ))
H̄m+1 = ā y
Fig. 1. Demonstration of forward and backward problems under analysis
(an inhomogeneous bianisotropic MM structure). (a) Recursive S11 and S21
parameters of the forward problem. (b) Recursive S12 and S22 parameters of
the backward problem.
extraction algorithm for the determination of electromagnetic
parameters of any homogeneous layer using two-sided recursive S-parameter expressions. We next performed two different
analyses (numerical and simulation) for the validation of
our algorithm by considering three different inhomogeneous
structures in Section IV. In this section, we also investigated the effect of random noise superposed on simulated
S-parameters on the performance of our extraction algorithm.
Finally, in Section V, we recapitulate some important points
about our extraction process and its application.
II. T HEORY
Fig. 1(a) and (b) shows the geometrical configurations for
the derivation of recursive forward (S11 and S21 ) and backward
(S12 and S22 ) S-parameters of a bianisotropic inhomogeneous
MM structure, respectively. The forward and backward wave
impedances, refractive index, and thickness of the mth layer
+ , Z − , n , and d , respectively. To make
are denoted by Z m
m
m
m
the analysis as general as possible, the inhomogeneous MM
structure is assumed to be surrounded by two different bianisotropic MM layers (Layers 0 and m + 1).
Without loss of generality, we assume that the
inhomogeneous structure in Fig. 1 is reciprocal (transmission
symmetric) and that a uniform plane wave propagating in
+z direction with polarization in +x direction impinges onto
the MM structure in Fig. 1(a), which can be separated into
m-layer piecewise homogeneous bianisotropic MM slabs.
Then, the expressions of electric and magnetic fields in the
sth layer (s = 0, 1, . . . , m) can be written for the exp(− j ωt)
(1)
Ē s = āx E si e+ j k0 ns (z−dt ) + E sr e− j k0 ns (z−dt )
i
r
E
E s + j k0 ns (z−dt )
H̄s = ā y
− −s e− j k0 ns (z−dt )
(2)
+e
Zs
Zs
i
E m+1
+
Z m+1
e+ j k0 nm+1 (z−(d1+d2 +···+dm )) .
(4)
(5)
In (1)–(5), the superscripts “i ” and “r ” denote the amplitudes
of incident and reflected waves; k0 is the free-space wavenumber; and n s , Z s+ , and Z s− are the refractive index and forward
and backward wave impedances of the sth layer.
Applying boundary conditions (continuity of tangential
components of electric and magnetic fields) at interfaces
(z = 0, d1 , d1 +d2 , . . . , d1 +d2 +· · ·+dm−1 ) and following the
procedure in [40], we obtain the following recursive relations
for forward S-parameters (l = 1, 2, . . . , m − 1):
(l ) + +
−
− +
+
Z l− S111 Z l1 Z l + Z l1 − Z l1 Z l − Z l1
1
(l)
(6)
S11 = +
(l
)
+
−
−
−
+
−
1
Zl
Tl2
S11 Z l1 Z l1 − Z l + Z l1 Z l1 + Z l
(l1 )
Z l+1 Z l−1 Z l+ + Z l− S21
1
1
(l)
(7)
S21 = +
−
+
(l
)
+
−
−
−
1
Tl
Zl
S Z Z −Z +Z Z +Z
11
l1
l1
l
l1
l1
l
where l1 = l − 1, l2 = l + 1, and
(l1 )
S11
=
Elr1
Eli1
(l1 )
S21
=
i
E m+1
Eli1
(0)
(0)
= S11 , S21
= S21
Tl = e+ j k0 nl dl S11
(8)
(9)
(l1 )
(l1 )
where S11
and S21
, defined at z = d1 +d2 +· · ·+dl−1 , denote
the total forward reflection and transmission S-parameters for
the structure involving the layers l, l + 1, . . . , m, respectively.
In addition, application of the boundary condition (continuity of tangential components of electric and magnetic fields)
at z = d1 + d2 + · · · + dm , we find
− Z+
+
r
Zm
− Zm
Em
(m)
S11 = i = + m+1
(10)
+
−
Em
Z m Z m+1 + Z m
+ +
i
−
Zm + Zm
E m+1
Z m+1
(m)
S21 =
(11)
=
+
+
− .
i
Em
Zm
+ Zm
Z m+1
Now, we assume that a uniform plane wave propagating in
the −z direction with polarization in +x direction impinges
onto the MM structure in Fig. 1(b). Then, applying the
same boundary conditions for the derivation recursive relations
for S11 and S21 at interfaces (z = d1 + d2 + · · · + dm ,
d1 + d2 + · · · + dm−1 , . . . , d1 ) and following the procedure
in [40], the recursive relations for the backward reflection and
transmission S-parameters are derived as:
(l )
− Z+ + Z− + Z+ Z− − Z−
S221 Z m
m2
m1
m2
m2
m1
(l)
2
S22 = ψ1
(12)
(l1 ) − +
+
+ −
+
S22 Z m 2 Z m 2 − Z m 1 + Z m 2 Z m 2 + Z m
1
+
− S (l1 )
Z m1 + Z m
(l)
12
1
S12 = ψ2
(13)
(l1 ) −
+
+
+ −
+
S22 Z m 2 Z m 2 − Z m 1 + Z m 2 Z m 2 + Z m
1
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HASAR et al.: DETERMINATION OF EFFECTIVE CONSTITUTIVE PARAMETERS OF INHOMOGENEOUS MMs
Fig. 2.
Extraction of wave impedances (Z u+ and Z u− ) and refractive
index (n u ) of the uth layer (1 ≤ u ≤ m) in the m layer inhomogeneous
structure in Fig. 1.
where m 1 = m − l + 1, m 2 = m − l + 2, and
ψ1 =
+
Zm
1
− 2
Zm
1 Tm 1
ψ2 =
+ Z−
Zm
2 m2
−
Zm
1 Tm 1
.
(14)
Application of the boundary conditions at z = 0 yields
Z 1+ Z 0− − Z 1−
E 1r
(m)
(15)
S22 = i = − +
Z 1 Z 1 + Z 0−
E1
Z 0− Z 1+ + Z 1−
E 0i
(m)
S12 = i = − +
.
(16)
Z 1 Z 1 + Z 0−
E1
We note that setting Z l+1 = Z l−1 = ηl1 , Z l+ = Z l− = ηl ,
(l)
+ = Z − = η , and Z + = Z − = η
Zm
m1
m 2 reduces S11 ,
m1
m2
m2
1
(l)
(l)
(l)
S21 , S22 , and S12 in (6), (7), (12), and (13) to the recursive
expressions of forward and backward reflection and transmission S-parameters in (10) and (13) [40] for an inhomogeneous
structure involving isotropic MM slabs.
III. E XTRACTION P ROCESS
Using the recursive expressions in (6)–(16), it is possible
to calculate intermediate S-parameters of a bianisotropic MM
(u 1 )
(u 1 )
, S21
,
layer in Fig. 1. For example, as shown in Fig. 2, S11
(m−u)
(m−u)
S12
, and S22
intermediate S-parameters can be used to
retrieve the wave impedances (Z u+ and Z u− ) and the refractive
index (n u ) of the uth layer (1 ≤ u ≤ m) in the inhomogeneous
structure with an m layer, as shown in Fig. 1. However, such
an extraction for the geometry in Fig. 2 requires the knowledge
of wave impedances and refractive indices of the first (from
left to right) u − 1 layers and the last u + 1 to m layers in the
inhomogeneous structure in Fig. 1, since S-parameters (S11 ,
S21 , S12 , and S22 ) of the whole inhomogeneous MM structure
in Fig. 1 are not sufficient to determine the electromagnetic
properties of each layer of this inhomogeneous MM structure
with m > 1. Therefore, we assume that the wave impedances
and refractive indices of the first u −1 layers and the last u +1
to m layers are known or extracted by a retrieval procedure,
to be discussed later.
In the extraction of Z u+ , Z u− , and n u of the uth layer
in Fig. 2, we propose the following steps. We assume without
+
−
= Z m+1
= Z air and
loss of generality that Z 0+ = Z 0− = Z m+1
n 0 = n m+1 = n air , where Z air and n air are the impedance and
refractive index of air, respectively.
3
1) The effective electromagnetic properties (wave
impedances and refractive index) of the first u − 1
layers and the last u + 1 to m layers are extracted by a
suitable retrieval procedure such as [17] or [24] using
simulated or measured S-parameters of each one of
these layers (immersed in air region). These parameters
will be used in the following steps in the determination
of intermediate S-parameters by recursive relations.
(0)
(0)
= S11 and S21
= S21 and continue if u = 1.
2) Use S11
Replace the integer index l with the dummy integer
index p (1 ≤ p ≤ m) in the recursive relations in (6)
( p)
( p)
and (7) and then iterate S11 and S21 for p < u. For
example, for m = 6 and u = 4, one should obtain
(1)
(2)
(3)
(0)
(1)
(2)
S11 , S11 , and S11 in terms of S11 , S11 , and S11 ,
( p)
respectively. A similar result will be obtained for S21 .
(m)
3) If u = m, implement the following. Calculate S11
and
(m)
(l−1)
(l−1)
S21 from (10) and (11) and express S11 and S21 in
(l)
the recursive relations in (6) and (7) in terms of S11 and
(l)
S21
. Then, replace the integer index l with the dummy
( p)
( p)
index p (1 ≤ p ≤ m) and iterate S11 and S21 for
u < p < m. For example, for m = 6 and u = 4, one
(5)
(6)
should obtain S11 in terms of S11 . Similar parameters
( p)
will be determined for S21 .
(0)
4) Use S22 = S22 and if u = m, continue with the
following steps. Replace the integer index l with the
dummy integer index p (1 ≤ p ≤ m) in the recursive
( p)
relation in (12). Then, iterate S22 for 1 ≤ p ≤ m 3 where
m 3 = m − u. For example, for m = 6 and u = 4, one
(1)
(2)
(0)
(1)
should obtain S22 and S22 in terms of S22 and S22 ,
respectively.
(m)
5) If u = 1, implement the following. Calculate S22
(l−1)
from (15) and express S22
in the recursive relation
(l)
in (12) in terms of S22 . Then, replace the integer index
( p)
l with the dummy index p (1 ≤ p ≤ m) and iterate S22
for m 4 < p ≤ m where m 4 = m − u + 2. For example,
(5)
for m = 6 and u = 4, one should obtain S22 in terms
(6)
of S22
.
(m )
6) By setting Tu21 S22 4 to zero for u = 1, and by setting
(u )
(u )
Tu22 S112 to zero and Tu22 S212 to one for u = m,
+
calculate Z u from
10 [(2 6 −3 5 )10 +(2 4 −1 5 )9 ]Z u+
+ [2(2 6 −3 5 )9 10 +(2 4 −1 5 )
3
× 8 10 +229 +(3 4 −1 6 )×210 ]Z u+
4
+ 39 [(2 4 −1 5 )8 +(3 4 −1 6 )10 ]
× Z u+ +[(2 4 −1 5 )28 +2(3 5 −2 6 )
× 8 9 +(3 4 −1 6 ) 8 10 +229 ]Z u+
2
+ 8 [(3 5 −2 6 )8 +(3 4 −1 6 )9 ] = 0
(17)
where
(u 1 ) +
(m 3 ) −
Z u 2 Z u+1 Z u−1
Z u 2 − S22
1 = 1 + S11
(m 3 )
(u 1 ) +
+ 1 + S22
Z u 1 − Z u−1 Z u+2 Z u−2
S11
(18)
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(u 1 ) + +
(m 3 ) −
2 = Z u−1 − S11
Z u 1 Z u 2 − S22
Z u2
(u )
(m )
3 = − 1 + S111 Z u+1 Z u−1 1 + S22 3 Z u+2 Z u−2
(u 2 )
4 = 1 + Tu22 S11
1−
(m 4 )
− 1 + Tu21 S22
5 =
1−
Z u+2
Z u−1
Z u+1
1−
(u )
(20)
(m 4 )
Z u+2
Tu21 S22
Z u+2
Z u−2
1−
(u 2 )
Z u−1 (21)
Tu22 S11
Z u−1
(m )
Tu21 S22 4
(22)
(u 2 )
(m 4 )
6 = − 1 + Tu22 S11
Z u+2 1 + Tu21 S22
Z u−1
(23)
7 =
Z u−2
Tu22 S112
(19)
(u 2 )
Z u+2 Tu 2 S21
(u )
Z u+1
Z u+1 Z u−1 S211
(u 1 ) 2 +2 −2 2
− 1 + S11
Z u 1 Z u 1 7
(u 1 ) −
(u 1 ) +
9 = 1 + S11
Z u 1 − S11
Z u 1 27 Z u+1 Z u−1
(u 2 )
− 1 + Tu22 S11
10 =
1−
Z u+2
Fig. 3.
Geometry of the analyzed inhomogeneous structures by CST
simulations. (a) Two-layer structure consisting of FR4 and SRR-wire MM
slab. (b) Three-layer structure consisting of SRR MM slab (Bian.), FR4, and
SRR-wire MM slab (Bian.). (c) Four-layer structure consisting of FR4, omega
MM slab (Bian.), marble, and omega-wire MM slab (Bian.).
(u 2 ) 2 +2
8 = 1 + Tu22 S11
Z u2
(u )
Tu2 S 2
Z u−2 2 11
1−
2
Z u+2 2 (u 2 )
Tu S
Z u−2 2 11
(24)
Z u+2 (25)
2
(u )
− Z u−1 − S111 Z u+1 27 .
(26)
Such a calculation can be performed by a suitable
numeric function such as Newton’s method [45] or
“roots” function of MATLAB. Once Z u+ is computed,
Z u− and Tu2 can be found from (A.1) and (A.11). Details
of the derivations of Z u+ , Z u− , and Tu can be found in
the Appendix. Finally, n u can be calculated from
−i ln(Tu2 ) + 2πm b
, m b = 0, ∓1, ∓2, . . . (27)
2k0 du
where m b denotes the branch index number. For
electrically small layer u, m b is essentially equal
to zero. However, for electrically large layer u, m b
attains various values. Correct value(s) of m b for the
layer u can be found by applying different techniques, such as the phase unwrapping method [23], [32],
the stepwise method [19], [24], or the improved NRW
algorithm [34].
7) Reiterate the steps 2 through 6 for any other medium
( p = u) in the inhomogeneous MM slab using newly
found electromagnetic parameters (Z u+ , Su− , and n u ) of
the uth layer.
8) Repeat the steps 1 through 7 for the layers whose
electromagnetic parameters are determined till the calculated electromagnetic parameters converge to the limit
of prespecified accuracy or till the end of a maximum
iteration number (Nmax ) set priori.
We want to highlight three important points regarding to
our extraction algorithm. First, the algorithm in [40] is suitable
for the extraction of electromagnetic properties of an inhomogeneous MM slab composed of isotropic layers, whereas our
extraction algorithm is applicable to electromagnetic parameter
nu =
retrieval of an inhomogeneous MM slab composed of either
isotropic or bianisotropic layers (or their combination). Second, while the algorithm in [40] works only in one direction
for electromagnetic parameter extraction, our algorithm is
two-sided searching for S11 from left to right and S22 from
right to left for the same goal. In other words, whereas the
(u−1)
(u−1)
(m−u)
and S21
(or S12
and
algorithm in [40] requires S11
(m−u)
S22
depending on iteration direction) for extraction of the
electromagnetic properties of the layer u in Fig. 2, our method
(u−1)
(m−u)
(u−1)
(m−u)
, S22
, and S21
(or S12
for isotropic
requires S11
materials) for the same purpose. Third, when m = u = 1 and
Z 0+ = Z 0− = Z 2+ = Z 2− = Z 0 are considered, our algorithm
reduces to the extraction method [17], [24] valid for only a
one-layer homogeneous bianisotropic MM slab.
IV. R ESULTS AND D ISCUSSION
In this section, we first validated and tested our proposed
method for three different inhomogeneous MM structures
composed of different MM slabs (split-ring-resonator (SRR),
SRR-wire, omega, and omega-wire). Then, we continued with
the effects of noise and iteration on extracted electromagnetic
parameters by our method.
A. Simulation Results
We applied the CST Microwave Studio with its time-domain
solver to obtain simulated S-parameters of the inhomogeneous
structures in Fig. 3(a)–(c). Simulation details are as follows.
Two waveguide ports were located on x y planes at z = 0 and
z = d1 + d2 + · · · + dm for the extraction of electromagnetic
properties of inhomogeneous MM structures, two of which
can show bianisotropic property. Perfect electric (magnetic)
conductors were assumed to be located on yz (x z) planes to
simulate uniform plane wave propagation in the +z direction
with polarization in the +x direction. Adaptive meshing
feature of the CST program was set active so that the CST
can arrange optimum meshing. Frequency range was set
to 2–18 GHz.
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HASAR et al.: DETERMINATION OF EFFECTIVE CONSTITUTIVE PARAMETERS OF INHOMOGENEOUS MMs
Fig. 4.
Extraction results for the two-layer inhomogeneous structure
[FR4 material and SRR-wire MM slab (Iso.)]. (a) Normalized (relative to air)
forward wave impedance of the SRR-wire MM slab. (b) Refractive index of
the SRR-wire MM slab. Magnitudes of (c) S11 and S22 and (d) S21 and phases
of (e) S11 and S22 and (f) S21 of the two-layer inhomogeneous structure
reconstructed from effective parameters (“Effective”) extracted by our method
and obtained from simulations (“Two-layer”). In the figures, “SM” and “PM”
refer to the stepwise method [24] and the proposed method.
We first considered the two-layer inhomogeneous structure
(m = 2) in Fig. 3(a) composed of an FR4 material
(d1 = 2.0 mm) and a composite SRR-wire MM slab. The
dimensions of this slab were set the same as those of the
study [40] in order to compare the results. The cell is cubic
with side d2 = 2.5 mm. The SRR is constructed over the
front face of the FR4 substrate with thickness 0.25 mm and
relative electrical permittivity εr1 = 4.4(1+ j 0.02) ( μr1 = 1).
For the SRR slab, the outer ring is 2.2 mm, the separation
distance between rings is 0.15 mm, the linewidth of each ring
is 0.2 mm, and the gap of both rings is 0.15 mm. Besides,
the wire with 0.2-mm linewidth extends in the x-direction at
the back of the FR4 substrate. Copper material with a thickness
of 35 μm is utilized to create SRR and wire strips. It is
known that this SRR-wire MM slab shows isotropic (Iso.)
property [11], [46] for the assumed plane wave propagation
in the z-direction with polarization vector in the x-direction,
i.e., S11 = S22 .
Applying the steps given in Section III, effective electromagnetic parameters of the SRR-wire MM slab (Iso.) of
this inhomogeneous structure were extracted by our algorithm for Nmax = 3 and with 127 968 hexahedral cells
used to discretize the computational domain. For comparison, these parameters were also retrieved by different extraction methods [11], [12], [17], [24], [25], [39] using
only the simulated S-parameters of the SRR-wire MM slab.
Because extracted electromagnetic properties by these methods are similar, only the results of the method in [24] are
presented in Fig. 4(a) and (b) for conciseness. Retrieved
z 2− is not demonstrated, since it is almost equal to z 2+ .
5
Fig. 5. Extracted wave impedances of the two-layer inhomogeneous structure
[FR4 material and SRR-wire MM slab (Iso.)]. (a) Normalized wave impedance
extracted by the methods in [11] and [12]. (b) Normalized wave impedance
extracted by the method in [39]. Normalized (c) forward and (d) backward
wave impedances extracted by the methods in [17] and [24]. Normalized
(e) forward and (f) backward wave impedances extracted by the method
in [25].
In addition, as shown in Fig. 4(c)–(f), we also compared
the simulated S-parameters of the two-layer inhomogeneous
structure with its S-parameters obtained from the effective parameters by our algorithm using the transfer matrix
method (TMM) from [43, expressions (5) and (6)]. The
following points are noted from dependences in Fig. 4(a)–(f).
First, normalized forward (and backward) wave impedance
z 2+ (and z 2− ) and the refractive index n 2 of this SRR-wire
MM slab extracted by our algorithm and the extraction
methods [11], [12], [17], [24], [25], [39] are in good agreement with one another except for some small differences.
We think that these differences arise from two main factors.
As a first factor, as noted in [40], there might be a coupling
between layers of the inhomogeneous structure, which in turn
can alter the electromagnetic properties of the SRR-wire (Iso.).
As a second factor, there might be some spatial effects as a
consequence of near-field boundary effects near the border
of the SRR-wire MM slab [30], [36], which can also modify its effective electromagnetic parameters. Second, as seen
in Fig. 4(c)–(f), both the magnitudes and phases of simulated
and reconstructed (by our extraction technique) S-parameters
of the SRR-wire MM slab are in good agreement with each
other except for some small discrepances over whole band,
validating our extraction method and showing its accurateness
for the extraction of electromagnetic properties of an isotropic
MM slab of an inhomogeneous MM structure.
We note that the methods in [11], [12], [17], [24], [25],
and [39] are applicable to retrieval of electromagnetic
properties of one homogeneous MM slab. It is instructive
to analyze the applicability of these methods for inhomogeneous MM slabs. For example, Fig. 5(a)–(f) shows
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6
the extracted wave impedances of the two-layer inhomogeneous structure (FR4 material and SRR-wire MM slab) with
thickness dt = d1 + d2 = 4.5 mm by different extraction
methods [11], [12], [17], [24], [25], [39]. Extracted refractive
index is not shown for simplicity. The following points are
noted from the dependences in Fig. 5. First, none of the extraction methods [11], [12], [17], [24], [25], [39] could retrieve
the correct wave impedances [see Fig. 4(a) and (b)] of the
SRR-wire slab. Second, although the SRR-wire slab demonstrates isotropic behavior [see Fig. 3(a)], extracted forward and
backward wave impedances by the methods in [17] and [24]
and the method in [25] significantly differ from each other
due to the presence of the FR4 material. Third, real parts of
the forward and backward wave impedances of the two-layer
inhomogeneous structure retrieved by the method in [25] could
result in nonphysical artifacts (e{z} ≤ 0) around 8 and
10 GHz.
Besides, we also employed the extraction algorithm [40] to
extract electromagnetic parameters of the SRR-wire MM slab
of the two-layer inhomogeneous structure in Fig. 3(a) because
it does not possess bianisotropy property for assumed plane
wave configuration. Electromagnetic parameters z 2+ , z 2− , and
n 2 extracted by the extraction algorithm [40] are so close to
those retrieved by our extraction algorithm in Fig. 4(a)–(c)
(so they are not presented here), demonstrating concurrency
between our algorithm and the algorithm in [40]. It is noted
that when the bianisotropy breaks down (z + = z − or
S11 = S22 ), our method and the method in [40] become very
similar.
As a second example, we investigated the three-layer
(m = 3) inhomogeneous structure in Fig. 3(b) composed of an
SRR MM slab (d1 = 2.5 mm), FR4 material (d2 = 2.0 mm),
and the SRR-wire MM slab (d3 = 2.5 mm). Here, the
SRR-wire MM slab has the same electrical and geometrical
properties of the SRR-wire MM slab in the previous example
(isotropic MM slab) except that the rings are rotated in the
+y direction by 90°. By this new orientation of the ring gaps,
as seen in Fig. 3(b), the magnetic field in the y-direction
(normal to the plane of the SRR) will produce currents and
thus electric dipoles in the x-direction due to gap. Additionally,
the electric field in the x-direction (normal to the slit axis) will
induce deposition of charges and thus magnetic dipole in the
y-direction. As such, this new topology of the SRR-wire MM
slab possesses bianisotropic (Bian.) property (z + = z − and
S11 = S22 ) for the wave propagation in the z-direction with
polarization vector in the x-direction [17], [37], [38], [46].
The SRR MM slab is identical to the SRR-wire MM slab
without the wire at the back of the FR4 substrate.
Effective electromagnetic parameters of the SRR-wire
(Bian.) of the three-layer inhomogeneous structure were
extracted by our algorithm for Nmax = 3 and with
153 216 hexahedral cells used to discretize the computational domain. For comparison, we also applied the extraction
methods in [17], [24], and [25] to inverse the electromagnetic properties of the SRR-wire MM slab (Bian.) using
its simulated S-parameters only. We note that the extraction methods [11], [12], [17], [24], [25], [39] were not tested
because they are not applicable for bianisotropic MM slabs.
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Fig. 6.
Extraction results for the three-layer inhomogeneous structure
[SRR (Bian.) MM slab, FR4 material, and SRR-wire MM slab (Bian.)].
(a) Normalized (relative to air) forward wave impedance of the SRR-wire
MM slab. (b) Normalized (relative to air) backward wave impedance of the
SRR-wire MM slab. (c) Refractive index of the SRR-wire MM slab. (d) Closer
view of the extracted m{z 3− }. Magnitudes of (e) S11 and S22 and (f) S21 .
Phases of (g) S11 and S22 and (h) S21 of the three-layer inhomogeneous
structure reconstructed from effective parameters (“Effective”) extracted by
our method and obtained from simulations (“Two-layer”). In the figures,
“LAO” and “PM” refer to the retrieval method [17] and the proposed method.
Since the methods in [17], [24], and [25] produce similar
results, only the results of the method in [17] are presented
in Fig. 6 for brevity. For example, Fig. 6(a)–(d) shows the
frequency dependences of z 3+ , z 3− , and n 3 of the SRR-wire
MM slab extracted from our algorithm and the method in [17].
The following two points are noted from the dependences
in Fig. 6(a)–(d). First, z 3+ and z 3− are different from each
other over the whole frequency band, demonstrating bianisotropic feature of the analyzed SRR-wire MM slab. Second,
the normalized wave impedances z 3+ and z 3− and the refractive
index n 3 of the SRR-wire (Bian.) MM slab obtained by
the sole extraction procedure [17] and by our algorithm are
generally in good agreement over the whole band except
for the two frequency regions where rapid variations are
notable ( f ∼
= 8.67 GHz where |S11 | attains its minimum
and f ∼
14.9
GHz where n 3 approaches zero). For exam=
ple, Fig. 6(d) shows the dependence of the imaginary part
of the extracted z 3− (m{z 3− }) over 14–16 GHz, indicating
some differences between extracted parameters. Specifically,
m{z 3− } values at f = 14.8 GHz extracted by the sole
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HASAR et al.: DETERMINATION OF EFFECTIVE CONSTITUTIVE PARAMETERS OF INHOMOGENEOUS MMs
extraction method [17] and the proposed algorithm are −5.70
and −5.14, respectively—showing an approximately 10%
variation. We think that the coupling between MM slabs as
well as the spatial effects around the borders of MM slabs are
the cause of discrepancy between extracted parameters by the
method in [17] and our algorithm.
We also analyzed and compared the simulated S-parameters
of the three-layer inhomogeneous structure with its
S-parameters reconstructed from the effective parameters by
our algorithm via the TMM method [43]. The results are
shown in Fig. 6(e)–(h). We note the following two points from
these figures. First, reconstructed individual S-parameters
(S11 , S21 , and S22 ) of this inhomogeneous structure by
our algorithm partly differ from its simulated S-parameters
specially around the resonance regions f ∼
= 8.67 and
f ∼
= 11.07 GHz. Second, the difference between reconstructed
and simulated individual S-parameters of this inhomogeneous
structure [see Fig. 3(b)] is overall greater than the difference
between reconstructed and simulated individual S-parameters
of the two-layer inhomogeneous structure [see Fig. 3(a)].
We think that there are two reasons for such an increase.
First, coupling between electric and magnetic fields for an
individual SRR (or SRR-wire) increases due to the rotation
of the gap axis of the SRR rings. Second, there might be
an additional coupling between the resonating MM slabs of
the inhomogeneous structure in Fig. 3(b). Finally, we note
from the dependences in Fig. 6(e)–(h) that our proposed
method is applicable for the extraction of the electromagnetic
properties of bianisotropic MM slabs of an inhomogeneous
MM structure.
In order to test the applicability of our proposed method
for different MM slabs, as a final example, we examined
the four-layer inhomogeneous structure in Fig. 3(c) composed
of an FR4 material (d1 = 5.0 mm), an omega MM slab
(Bian.), a PVC material (d3 = 5.0 mm, εr3 = 2.71(1 + 0.08),
and μr3 = 1), and an omega-wire MM slab (Bian.). The
geometrical parameters and the electromagnetic properties of
both omega and omega-wire MM slabs are similar to those
in [24]. The cell of the omega MM slab has a mean radius
of 1.19 mm, a width of 0.45 mm, and the tail length of 1.8 mm,
and placed over the front face of the FR4 substrate with
a thickness of 1.6 mm. The wire with 1.44-mm linewidth
extends in the x-direction at the back of the FR4 substrate.
Copper material with a thickness of 35 μm is utilized to
create omega patterns and wire strips. Both the omega and
omega-wire slabs are in cubical form with a side of d2 =
d4 = 5.0 mm, exhibiting bianisotropic behavior since electric
field is parallel with the tails and magnetic field is normal to
the plane of the omega-shape [24], [47].
Effective electromagnetic parameters of the omega MM
slab (Bian.) of the four-layer inhomogeneous structure were
retrieved by our algorithm for Nmax = 3 and with
632 016 hexahedral cells used to discretize the computational
domain. For comparison, we applied the extraction methods
in [17], [24], and [25] to inverse the electromagnetic properties of the omega MM slab (Bian.) using its simulated
S-parameters only. Only the results of the method in [17]
are presented in Fig. 7 for brevity, because the methods
7
Fig. 7. Extraction results for the four-layer inhomogeneous structure [FR4,
omega MM slab (Bian.), PVC sample, and omega-wire MM slab (Bian.)].
(a) Normalized (relative to air) forward wave impedance of the omega MM
slab. (b) Refractive index of the omega MM slab. Magnitudes of (c) S11 and
S21 and phases of (d) S11 and S21 of the four-layer inhomogeneous structure
reconstructed from effective parameters (“Effective”) extracted by our method
and obtained from simulations (“Two-layer”). In the figures, “LAO” and “PM”
refer to the retrieval method [17] and the proposed method.
Fig. 8. Results for the real part of the normalized forward wave impedance
of the omega MM slab of the four-layer inhomogeneous structure [FR4,
omega MM slab (Bian.), PVC sample, and omega-wire MM slab (Bian.)].
(a) Results for our method and the method in [40] (denoted as “Shi-Liang”)
for Nmax = 3. (b) Results for our method and the method in [40] (denoted as
“Shi-Liang”) for Nmax = 4. Extraction results by the method in [17] (denoted
as “LAO”) are also shown here for comparison purposes.
in [17], [24], and [25] output similar results. For instance,
Fig. 7(a) and (b) shows the frequency dependences of z 2+ and
n 2 of the omega MM slab extracted from our algorithm and
the method in [17]. For brevity, the frequency dependence
of z 2− is not presented in Fig. 7. Good agreement between
retrieved electromagnetic parameters z 2+ and n 2 of the omega
MM slab by our method and the methods in [17], [24],
and [25] is observed in Fig. 7(a) and (b). Additionally,
we also compared the simulated S-parameters of the four-layer
inhomogeneous structure with its S-parameters obtained from
the effective parameters by our algorithm using the TMM
method [43], as shown in Fig. 7(c) and (d). We note from
Figs. 5(e)–(h) and 7(c) and (d) that our method is applicable
for the extraction of the electromagnetic properties of different bianisotropic MM slabs of various inhomogeneous MM
structures.
In addition to the application of the extraction
methods [17], [24], [25] and our method for obtaining
the electromagnetic parameters of the omega MM slab
of the four-layer inhomogeneous structure in Fig. 3(c),
we also applied the method in [40] for the same purpose.
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8
Fig. 8(a) and (b) shows the retrieved real part of z 2+ of the
omega MM slab for different Nmax values. Extracted z 2−
and n 2 are not shown here for conciseness. Results from
the extraction method [17] are also shown in Fig. 8 for
comparison. While simulated S-parameters of the omega
MM slab are used in the application of the method in [17],
simulated S-parameters of the four-layer structure are used in
the application of the method in [40] and our method. From
the dependences in Fig. 8(a) and (b), we note two points.
First, extracted real parts of z 2+ by our method are more closer
than the retrieved real parts of z 2+ by the method in [40] to
the extracted real part of z 2+ by the method in [17] for two
different values of Nmax . This is because while our proposed
method is applicable for the extraction of electromagnetic
properties of bianisotropic materials in inhomogeneous
structures, the method in [40] is only applicable for the
extraction of electromagnetic properties of isotropic materials
of inhomogeneous structures. Second, while extracted real
parts of z 2+ by the method in [40] change for different Nmax
values, extracted real parts of z 2+ by our method do not
change much for different Nmax values. This is because
our proposed method uses a two-sided propagation and the
method in [40] uses a one-sided propagation in extracting
electromagnetic parameters of materials.
B. Analysis of Noise Effect and Iterative Error
Up to now, we have applied our algorithm for the extraction
of electromagnetic properties of different inhomogeneous
structures without focusing on the effects of noise and iterative error. However, in a real measurement environment,
measured S-parameters have some errors arising from the
limits of the measuring instrument (e.g., a vector network
analyzer) such as finite directivity and dynamic range, as well
as from limited accuracy of calibration kits (such as thrureflect-line [48]) used for the calibration of the measuring
instrument. Therefore, their effects on our proposed algorithm
should be examined to assess the sensitivity and endurance
of our algorithm. Toward this end, in line with the extraction
algorithm [40], we superposed a random noise to the simulated S-parameters of the analyzed inhomogeneus structures,
then applied our extraction algorithm, and finally compared
these simulated S-parameters with S-parameters reconstructed
by our method using extracted effective parameters via the
TMM method [43]. For example, Fig. 9(a)–(d) shows the
extracted electromagnetic parameters (z 3+ , z 3− , and n 3 ) of the
SRR-wire MM slab (Bian.) of the three-layer inhomogeneous
structure in Fig. 3(b). Before the application of our algorithm,
a normally distributed random noise with the mean value
ρ = 0 and the standard deviation σ = 0.050 was applied
to both the real and imaginary parts of the simulated S11 of
this inhomogeneous structure over the frequency ranges 8–9
and 14.5–15.5 GHz, where the sharp variations of z 3+ and
z 3− are observable [see Fig. 6(a) and (b)]. It is seen from
Fig. 9(a)–(d) that electromagnetic parameters extracted by our
extraction algorithm still follow the electromagnetic properties
retrieved by the sole extraction method [24] (using simulated S-parameters of the SRR-wire MM slab (Bian.) only)
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Fig. 9. Extraction results for the three-layer inhomogeneous structure [SRR
MM slab (Bian.), FR4 material, and SRR-wire MM slab (Bian.)] in Fig. 3(b)
when the normally distributed random noise (mean value ρ = 0 and standard
deviation σ = 0.05) is added to S11 of the inhomogeneous structure
over 8–9- and 14.5–15.5-GHz ranges. (a) Normalized (relative to air) forward
wave impedance of the SRR-wire MM slab. (b) Normalized (relative to air)
backward wave impedance of the SRR-wire MM slab. (c) Refractive index of
the SRR-wire MM slab. (d) Closer view of the extracted m{z 3− }. Magnitudes
of (e) S11 and S21 and (f) S21 of the inhomogeneous structure reconstructed
from effective parameters (“Effective”) extracted by our method and obtained
from simulations (“Two-layer”). In the figures, “SW” and “PM” refer to the
stepwise method [24] and the proposed method.
although some random noise is applied over the S11 parameter.
However, addition of this noise over simulated S11 increases
the difference between electromagnetic parameters extracted
by our algorithm and by the sole extraction method [24].
For example, as noted from Figs. 6(d) and 9(d), m{z 3− }
values at f = 14.8 GHz extracted by the sole extraction
method [24] and the proposed algorithm are −5.73 and −5.14,
respectively—showing an around 12% variation, as compared
to 10% variation for the dependences obtained by the same
methods in Fig. 6(d).
On the other hand, we performed some numerical analysis
for understanding the effect of iterative errors on electromagnetic properties extracted by our algorithm. As noted
in [40], round-off error, the number of iterations, and the
electromagnetic parameters of layers initially extracted by the
methods in [11], [12], [17], [24], [25], and [39] (refer to the
first step of our algorithm given in Section III) may influence
the performance of our algorithm. Among these terms, we will
focus on the effect of the number of iterations. The interested
reader can refer to [40] for the analysis of other terms. Here,
we considered a three-layer (m = 3) inhomogeneous structure
composed of a bianisotropic MM slab (d1 = 6.0 mm), FR4
material (d2 = 10.0 mm), and Marble material (d3 = 5.0
mm) with relative complex permittivity εr3 = 8.3. For the
synthesis of the bianisotropic MM slab, we applied the Lorentz
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HASAR et al.: DETERMINATION OF EFFECTIVE CONSTITUTIVE PARAMETERS OF INHOMOGENEOUS MMs
9
electromagnetic parameters for the close Nmax values is within
the limit of accuracy set priori.
V. C ONCLUSION
Fig. 10.
Extraction results for the synthesized inhomogeneous structure
[bianisotropic MM slab (d1 = 6.0 mm), FR4 material (d2 = 10.0 mm), and
marble (d3 = 5.0 mm)] for different iteration numbers Nmax . (a) Real and
(b) imaginary parts of the normalized (relative to air) forward wave impedance
of the bianisotropic MM slab. In the figures, “SW” and “PM” refer to the
stepwise method [24] and proposed method.
dispersion model [13]. According to this model, εr1 , μr1 , and
ξr1 of this slab are calculated from
Fe f 2
− f e2 + j δe f
Fm f 2
μr1 = 1 − 2
f − f m2 + j δm f
Fξ f 2
ξ1 = 1 − 2
f − f ξ2 + j δξ f
εr1 = 1 −
f2
(28)
(29)
(30)
where f e , f m , and fξ are the electric, magnetic, and magnetoelectric resonance frequencies; f is the operating frequency;
δe , δm , and δξ are the electric, magnetic, and magnetoelectric
damping frequencies; and Fe , Fm , and Fξ are the coefficients
depending on structure of the material [13]. In our analysis,
Fe = 0.4, Fm = 0.4, Fξ = 0.15, f e = 6.0 GHz,
f m = 5.0 GHz, f ξ = 5.0 GHz, δe = 1.0 GHz, δm = 1.0 GHz,
and δξ = 1.0 GHz were utilized for the frequency range
1–10 GHz, producing a passive material with e{z 1+ } ≥ 0,
e{z 1− } ≥ 0, and m{n 1 } ≥ 0 [see Fig. 10(a) and (b)]. After
we determined forward and backward wave impedances and
refractive index of each material using [24, expression (3)].
For FR4 and marble samples, their magnetoelectric coupling
coefficients (ξ2 and ξ3 ) and relative permeabilities (μ2 and μ3 )
were set to zero and one, respectively. Next, S-parameters
of each material were computed from [24, expressions (1)
and (2)]. Finally, S-parameters of the inhomogeneous structure
were determined by using the TMM method [43].
Fig. 10(a) and (b) shows the frequency dependence of the
extracted real and imaginary parts of z 1+ of the bianisotropic
MM slab of the synthesized inhomogeneous structure. It is
seen from Fig. 10(a) and (b) that while z 1+ extracted by our
algorithm for Nmax = 40 follows the z 1+ extracted by the sole
extraction method [24], it is not true for Nmax = 80, especially
for frequencies between 4 and 6 GHz, where sharp variation
of electromagnetic parameters is notable. The reason for this
is that our proposed algorithm, as the iterative algorithm
proposed in [40], is iterative, and thus its accuracy is affected
by the accuracy of electromagnetic parameters of the layers
determined before each iteration process, as discussed in the
first step in our algorithm in Section III. Therefore, the value
of Nmax should be first set to a smaller number around the
resonance region of MM slabs and then can be increased
gradually to a point where the difference between extracted
We have proposed a recursive retrieval procedure for the
accurate determination of electromagnetic properties of bianisotropic inhomogeneous MM slabs using measured/simulated
S-parameters. For validation and accuracy analysis of our
proposed method, we performed simulation analysis by considering a total of three different inhomogeneous structures.
From this analysis, we noted the following points. First, sole
extraction methods do not accurately extract electromagnetic
properties of inhomogeneous materials, since they are applicable for homogeneous materials only. Second, the extraction
method applicable for isotropic inhomogeneous structures
fails to extract correct electromagnetic parameters of bianisotropic MM slabs constituting the analyzed inhomogeneous
structures. Third, extracted electromagnetic parameters of
isotropic/bianisotropic MM slab by our method slightly differ
from those obtained by the sole extraction methods (using
simulated S-parameters of the isotropic/bianisotropic MM slab
only). We think that this difference stems from a coupling
effect between the layers of the analyzed inhomogeneous
structure, spatial effects as a consequence of near field boundary effects, and coupling between electric and magnetic fields
(bianisotropic MM slabs). Fourth, while the accuracy of our
extraction method is not much altered by a small change in
the number of maximum iteration number (from Nmax = 3
to Nmax = 4), the accuracy of the extraction method feasible
for isotropic inhomogeneous MM slabs greatly suffers from
this change. This difference comes from the nature of the
algorithms of our method (two-sided propagation) and the
method for isotropic inhomogeneous MM slabs (one-sided
propagation). In addition to validation and accuracy analysis,
we also analyzed the effects of noise and iterative error by
using the results obtained from our algorithm and the method
existing in the literature. We observed that our proposed
algorithm is robust in producing accurate electromagnetic
parameters even in the presence of an additional noise effect.
Besides, we also observed that by increasing maximum iteration number considerably (from Nmax = 40 to Nmax = 80 for
the analysis of the synthesized inhomogeneous MM structure),
the accuracy of our iterative algorithm may decrease due to
any inaccurate determination of electromagnetic parameters of
the layers before each iteration process. Therefore, the value
of iteration numbers should be first set to a smaller number
around the resonance region of MM slabs and then can be
increased gradually to a point where the difference between
extracted electromagnetic parameters for the close two iteration number values is within the limit of accuracy set priori.
A PPENDIX
From the S11 expressions obtained from steps 2 and 3 in
Section III, one can determine Tu2 in terms of Z u+ and Z u−
in (A.1)–(A.5). In a similar manner, from the S22 expressions
obtained in steps 4 and 5 in Section III, one can obtain Tu2 in
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10
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES
terms of Z u+ and Z u− in (A.1) and (A.6)–(A.9)
2
χ1 χ2 2 χ5 χ6
Tu S =
Tu S =
11
22
χ3 χ4
χ7 χ8
(A.1)
where
Z u+
(u )
(u )
χ1 = 1 + Tu22 S112 Z u+2 + 1 − −2 Tu22 S112 Z u−
Z u2
(u 1 ) +
(u 1 )
Z u 1 − Z u−1 Z u+
χ2 = 1 + S11
Z u+1 Z u−1 + S11
(u )
χ3 = 1 + Tu22 S112 Z u+2 − 1 −
Z u+2 2 (u 2 )
Tu S
Z u−2 2 11
Z u+
(u 1 ) +
(u 1 )
Z u+1 Z u−1 − S11
Z u 1 − Z u−1 Z u−
χ4 = 1 + S11
χ5 = 1 +
χ6 = 1 +
χ7 = 1 +
χ8 = 1 +
Z u−
(m )
(m )
Tu21 S22 2 Z u−1 + 1 − +1 Tu21 S22 2 Z u+
Z u1
(m 1 ) −
(m 1 )
Z u+2 Z u−2 + S22
S22
Z u 2 − Z u+2 Z u−
Z u−
(m 2 )
(m 2 )
Z u−
Tu21 S22
Z u−1 − 1 − +1 Tu21 S22
Z u1
(m )
(m )
S22 1 Z u+2 Z u−2 − S22 1 Z u−2 − Z u+2 Z u+ .
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
(A.9)
Equating both Tu expressions in (A.1), one can find
4 Z u− + 5 Z u− + 6
2
1 Z u− + 2 Z u− + 3
2
−4 Z u+ + 5 Z u+ + 6
2
=
−1 Z u+ + 2 Z u+ + 3
2
(A.10)
which expresses the relation between Z u− and Z u+ [and electromagnetic parameters of other layers ( p = u)], where 1 ,
2 , 3 , 4 , 5 , and 6 are given in (18)–(23).
Then, from the S21 expressions obtained from steps 2 and 3
in Section III, Z u− is written by Z u+ and electromagnetic
parameters of other layers ( p = u) after substituting Tu2 for
S11 (or S22 ) in (A.1) into (A.10) as
Z u− =
8 + 9 Z u+
9 + 10 Z u+
(A.11)
where 8 , 9 , and 10 are presented in (24)–(26). Finally,
substituting (A.11) into (A.10), the metric function F(Z u+ )
in (17) is derived.
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Ugur Cem Hasar (M’00) received the B.Sc. and
M.Sc. degrees (Hons.) in electrical and electronics engineering from Cukurova University, Adana,
Turkey, in 2000 and 2002, respectively, and the
Ph.D. degree (Hons.) in electrical and computer
engineering from The State University of New York
at Binghamton, Binghamton, NY, USA, in 2008.
From 2000 to 2005, he was a Research and Teaching Assistant with the Department of Electrical and
Electronics Engineering, Cukurova University. From
2005 to 2008, he was a Research Assistant with the
Department of Electrical and Electronics Engineering, Ataturk University,
Erzurum, Turkey, where he was an Assistant Professor from 2009 to 2011 and
also an Associate Professor from 2011 to 2013. Since 2017, he has been
a full-time Professor with the Department of Electrical and Electronics
Engineering, Gaziantep University, Gaziantep, Turkey. His current research
interests include nondestructive testing and evaluation of materials using
microwaves, novel calibration-dependent, calibration-independent techniques
for the electrical and physical characterization of conventional materials
at microwaves, millimeter waves, and THz frequencies, high-temperature
packaging for high power density applications, porous silicon-based devices
and their applications, and metamaterials.
Gul Buldu received the B.Sc. degree (Hons.) from Mustafa Kemal University,
Hatay, Turkey, in 2008, and the M.Sc. degree from Gaziantep University,
Gaziantep, Turkey, in 2017.
She is also a Scholar with the Scientific and Technological Research
Council of Turkey under the TUBITAK Project 112R032. Her current research
interests include the characterization of materials by calibration-independent
methods and metamaterials.
Yunus Kaya received the B.Sc. degree in electrical and electronics engineering and B.Sc. degree
in mechanical engineering from Ataturk University, Erzurum, Turkey, in 2011 and 2012, respectively. He is currently pursuing the Ph.D. degree
in electrical and electronics engineering at Ataturk
University.
Since 2013, he has been an Instructor with
the Department of Electricity and Energy, Bayburt
University, Bayburt, Turkey. His research interests
include characterization of materials by microwaves
and metamaterials.
Gokhan Ozturk received the B.Sc. degree from
Fırat University, Elazığ, Turkey, in 2009, and the
M.Sc. degree from Ataturk University, Erzurum,
Turkey, in 2014, all in electrical and electronics
engineering.
He was a Research Assistant with the Department of Electrical and Electronics Engineering, Iğdır
University, Iğdır, Turkey, from 2010 to 2012, and
Kafkas University, Kars, Turkey, in 2012, respectively. Since 2012, he has been a Research Assistant
with the Department of Electrical and Electronics
Engineering, Ataturk University. His current research interests include the
characterization of material by microwaves, metamaterials, and numerical
methods in electromagnetics.