Consider a magnetized ferrite-wire waveguide structure situated between two half free spaces. Ferrites
to provide negative permeability and wire array to provide negative permittivity. The structure form left -
handed material (LHM) with negative refractive index. The transmission of electromagnetic waves
through the structure is investigated theoretically. Maxwell's equations are used to determine the electric
and magnetic fields of the incident waves at each layer. Snell's law is applied and the boundary conditions
are imposed at each layer interface to calculate the reflected and transmitted powers of the structure. Numerical
results are illustrated to show the effect of frequency, applied magnetic fields, angle of incidence
and LHM thickness on the mentioned powers. The analyzed results show that the transmission is very
good when the permeability and permittivity of the structure are both simultaneously negative. The frequency
band corresponding to this transmission can be tuned by changing the applied magnetic fields. The
obtained results are in agreement with the law of conservation of energy.
When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/2489

Shabat, Mohammed M.

Sid-Ahmed, Mohammed O.

Ubeid, Muin F.

SocioEconomic Challenges Journal (Sumy State University)

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ 
Vol. 4 No 1, 01009(4pp) (2012) Том 4 № 1, 01009(4cc) (2012) 
 
 
2077-6772/2012/4(1)01009(4) 01009-1  2012 Sumy State University 
Numerical Study of Negative-Refractive Index Ferrite Waveguide 
 
Muin F. Ubeid1,*, Mohammed M. Shabat1, Mohammed O. Sid-Ahmed2 
 
1 Department of Physics, Faculty of Science, Islamic University of Gaza, P.O. Box 108, Gaza,  
Gaza Strip, Palestinian Authority 
2 Department of Physics, Faculty of Science, Sudan University of Science and Technology,  
The Republic of the Sudan 
 
(Received 22 October 2011; revised manuscript received 26 February 2012; published online 14 March 2012) 
 
Consider a magnetized ferrite-wire waveguide structure situated between two half free spaces. Ferrites 
to provide negative permeability and wire array to provide negative permittivity. The structure form left-
handed material (LHM) with negative refractive index. The transmission of electromagnetic waves 
through the structure is investigated theoretically. Maxwell's equations are used to determine the electric 
and magnetic fields of the incident waves at each layer. Snell's law is applied and the boundary conditions 
are imposed at each layer interface to calculate the reflected and transmitted powers of the structure. Nu-
merical results are illustrated to show the effect of frequency, applied magnetic fields, angle of incidence 
and LHM thickness on the mentioned powers. The analyzed results show that the transmission is very 
good when the permeability and permittivity of the structure are both simultaneously negative. The fre-
quency band corresponding to this transmission can be tuned by changing the applied magnetic fields. The 
obtained results are in agreement with the law of conservation of energy. 
 
Keywords: Applied magnetic fields, Electromagnetic waves, Ferromagnetic material, Frequency, Reflect-
ed power, Transmitted power. 
 
 PACS numbers: 42.25Bs, 42.25.Dd, 42.25.Gy 
 
 
______________ 
* mubeid@mail.iugaza.edu 
1. INTRODUCTION 
 
Metamaterials (sometimes termed left-handed ma-
terials (LHMs)) are materials whose both permittivity  
and permeability  are negative and consequently have 
negative index of refraction. These materials are artifi-
cial and theoretically discussed first by Veselago [1] 
over 40 years ago. The first realization of such materi-
als, consisting of split-ring resonators (SRRs) and con-
tinuous wires, was first introduced by Pendry [2, 3]. 
Regular materials are materials with positive  and  
and termed right handed materials (RHMs). 
Magnetized ferrite is an additional alternative to 
SRR to provide negative permeability. Consequently, a 
LHM can be achieved by inserting a periodic continu-
ous wires into the ferrite material [4-7].Both  and  of 
this LHM have tunable properties by changing the ap-
plied bias magnetic fields and both exhibit negative 
values at certain frequency band. Y. He et al. [8] have 
studied the role of ferrites in negative index metamate-
rials. M. Augustine et al. [9] have formulated a theoret-
ical analysis of ferrite-superconductor layered struc-
tures. H. Zhao et al. [10] have studied a magnetotuna-
ble left-handed material consisting of yttrium iron gar-
net slab and metallic wires. R.X. Wu [11] has shown 
the effect of negative refraction index in periodic metal-
ferrite film composite. F.J. Rachford et al. [12] have 
performed simulations of ferrite-dielectric-wire compo-
site negative index materials. Q.X. Chu et al. [13] have 
shown a novel left-handed metamaterial with wire ar-
ray in a ferrite-filled rectangular waveguide.  
In this paper we consider a magnetized ferrite-wire 
structure inserted in vacuum. A plane polarized wave 
is obliquely incident on it.  and  of the structure can 
easily be tuned by changing the applied magnetic fields 
and negative refractive index of it is achieved at a cer-
tain frequency band to make the structure as LHM. 
Maxwell's equations are used to determine the electric 
and magnetic fields in each region. Then, Snell's law is 
applied and the boundary conditions of the fields are 
imposed at each interface to obtain a number of equa-
tions with unknown parameters. The equations are 
solved for the unknown parameters to calculate the 
reflection and transmission coefficients. These coeffi-
cients are used to determine the reflected and trans-
mitted powers of the structure. The effect of many pa-
rameters like frequency, applied magnetic fields and 
angle of incidence etc. on the mentioned powers are 
studied in detail. It is demonstrated that, if both ε and 
μ of the ferrite-wire structure are both positive or both 
negative, then the electromagnetic waves can propa-
gate through it. On the other hand if either  or  of the 
structure is negative, the incident radiation will be 
reflected and no transmission of the waves will be 
through the structure. Moreover the frequency band of 
zero or nonzero transmission of the waves can be tuned 
by changing the applied magnetic fields due to the ten-
ability of ε and μ of the structure. To check the validity 
of the performed computations the law of conservation 
of energy given by [14, 15 ] is satisfied for all examples. 
 
2. THEORY 
 
Consider a rectangular ferrite-wire waveguide 
structure located between two half free spaces and a dc 
applied magnetic field acts on it along the X axis [4, 5]. 
A plane polarized wave is incident on the plane y = 0 at 
angle  relative to the normal to the boundary (Fig.1). 
The effective permeability the ferrite is given by [4, 5]: 
 
 
2 2
f , (2.1) 
 MUIN F. UBEID, ET AL. J. NANO- ELECTRON. PHYS. 4, 01009 (2012) 
 
 
01009-2 
where 0
2 2
0
( )
1
( )
m i
i
, 
2 2
0( )
m
i
,   
 
0 B , m M ,  is the gyromagnetic ratio, B is 
the intensity of the applied magnetic field, M is the 
saturation magnetization, ω is the angular frequency 
and i = 1 . The effective permittivity of the structure 
is considered to be [4, 5]: 
 
 
6
0 0[ (1.57 10 )(3.18 .413 )]
eff
f r
eff fi
 (2.2) 
 
Where 0 and 0 are the permittivity and permeabil-
ity of free space respectively, r is the relative permit-
tivity of ferrite material and eff is the effective conduc-
tivity of wire arrays.  
 
 
 
Fig. 2.1 – Oblique incidence of electromagnetic wave on a 
ferrite-wire structure embedded in vacuum 
 
The transverse components of the electric and mag-
netic fields in each region of Fig. 2.1 are [9, 16]: 
Region 1: 
 
 ( )1 ( )oy oy oz
ik y ik y i k z t
xE Ae Be e  (2.3) 
 ( )1
0
1
( )oy oy ozik y ik y i k z tz oy oyH Ak e Bk e e  (2.4) 
 
Region 2: 
 
 ( )2 ( )fy fy fz
ik y ik y i k z t
xE Ce De e  (2.5) 
 
 ( )2
1
( )fy fy fzik y ik y i k z tz c bH Ck e Dk e e  (2.6) 
 
With: 
 
fz fy
c
f
k k
k , 
fz fy
b
f
k k
k , 
2 2
i
 (2.7) 
 
Region 3: 
 
 ( )3 oy oz
ik y i k z t
xE Fe e   (2.8) 
 ( )3
0
1
( )oy ozik y i k z tz oyH Fk e e  (2.9) 
 
Where A, B, C, D, F are the amplitudes of the for-
ward and backward traveling waves. k0  /c is the free 
space wave vector, kf  nfω/c is the wave vector inside 
the  slab and 0 0f f fn  is the refractive index of it. 
 sinoz fz ok k k  (Snell's Law) (2.10) 
 
 21 sinoyk
c
, 2 2sinfy fk n
c
 (2.11) 
 
Matching the boundary conditions for the trans-
verse field components at each layer interface, that is 
at y  0, 1 2x xE E  and 1 2z zH H , at y  a 2 3x xE E  
and 2 3z zH H . This yields the following equations [16]: 
 
 A + B  C + D (2.12) 
 
 
0
( )
oy
c b
k
A B Ck Dk   (2.13) 
 
 fy fy oyik a ik a ik aCe De Fe  (2.14) 
 
  
0
fy fy oy
oyik a ik a ik a
c b
k
Ck e Dk e Fe  (2.15) 
 
Letting A  1 and solving these four equations for the 
unknown parameters enables us to calculate the reflec-
tion and transmission coefficients B and F [16]. The re-
flected and transmitted powers are defined as [16]: 
 
 R  BB* (2.16) 
 
 T  FF* (2.17) 
 
Where B* and F* are the complex conjugate of B and 
F respectively. 
 
3. NUMERICAL RESULTS AND DISCUSSION  
 
In this section the reflected and transmitted powers 
of the structure are calculated numerically as a func-
tion of frequency, angle of incidence, applied magnetic 
fields and thickness of ferrite. In our method we have 
used the parameters as in [17]: M  4398 G, 
  2.8 MHz/Oe,   0.006, r  4. Three values of B are 
selected to be B  500 Oe, 750 Oe, and 1000 Oe. The 
central frequency is selected to be 8.6 GHz. This fre-
quency is chosen such that f and f are both simulta-
neously negative for all values of the applied magnetic 
fields. The slab thickness is assumed to be one half-
wavelength long at the central frequency. 
f ( ) with real part Re ( f) and f( ) with real Re( f) 
are plotted as a function of frequency  in Fig. 2.2 for 
three values of the applied magnetic fields ( B  500 Oe, 
B  750 Oe, B  1000 Oe). As shown from the figure the 
frequency range in which Re( f) and Re( f) are negative 
can be changed with the applied magnetic field [8]. In 
the case of B  500 Oe, f and f are both negative in the 
frequency range of (6.4-10.8 GHz) and consequently the 
ferrite-wire structure exhibits negative refractive index 
(i.e. LHM) in that range. For B  750 Oe and 
B  1000 Oe the ferrite-wire structure exhibits LHM in 
the frequency ranges of (7.2-11 GHz) and (8.2-11.2 GHz) 
respectively. 
The transmitted and reflected powers as a function 
of frequency are calculated in Fig. 3.2. for   30  and 
a  /2. The frequency is changed between 2 GHz and 
12 GHz, because the simultaneously negative values of 
Re( f) and Re( f) under the three values of B can be 
 NUMERICAL STUDY OF NEGATIVE-REFRACTIVE INDEX… J. NANO- ELECTRON. PHYS. 4, 01009 (2012) 
 
 
01009-3 
-60
-50
-40
-30
-20
-10
0
10
20
30
40
50
60
2 3 4 5 6 7 8 9 10 11 12
Frequency (GHz)
R
e
a
l 
p
e
rm
e
a
b
il
it
y
 a
n
d
 p
e
rm
it
ti
v
it
y Re (μf)     
B = 500 Oe
B = 750 Oe
B = 1000 Oe
Re (εf) 
B = 500 Oe
B = 750 Oe
B = 1000 Oe
 
Fig. 3.1 – Calculated effective permeability and permittivity 
under different applied magnetic fields B  500 Oe, B  750 Oe 
nd B  1000 Oe 
 
realized in this range. In this example it can be seen 
that, if both f and f are both positive or both negative, 
the transmission is very good and the electromagnetic 
wave can propagate through the ferrite-wire structure. 
This is because the propagation constant of the waves 
in the structure is real ( 2 2 2f f fk c ). If either f or 
f is negative, the propagation constant is imaginary and 
then the incident radiation will be reflected and no 
transmission of the waves will be through the structure. 
The symmetry of nonzero transmission shifts in frequen-
cy to the right side when the applied field increased. 
This is due to the tunability of the LHM which arises 
from the variation of the applied magnetic fields. 
0
0.2
0.4
0.6
0.8
1
2 4 6 8 10 12
Frequency (GHz)
T
ra
n
s
m
it
te
d
 P
o
w
e
r
B = 500 Oe
B = 750 Oe
B = 1000 Oe
 
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12
Frequency (GHz)
R
e
fl
e
c
te
d
 P
o
w
e
r
B = 500 Oe
B = 750 Oe
B = 1000 Oe
 
Fig. 3.2 – The transmitted and reflected powers against fre-
quency when the applied magnetic field changes B  500 Oe, 
B  750 Oe, and B  1000 Oe 
 
Fig. 3.3 shows the transmitted and reflected powers 
as a function of frequency when the incidence angle 
changes (   0 ,   30 ,   50 ) for B  500 Oe. The 
frequency range and thickness of ferrite is the same as 
in the last example. It can be seen that the transmitted 
power is decreasing while the reflected power increas-
ing with the angle of incidence. Moreover they show 
oscillatory behavior for   30  and 50  while they show 
nearly no ripples for   0 . 
0
0.2
0.4
0.6
0.8
1
2 4 6 8 10 12
Frequency (GHz)
T
ra
n
s
m
it
te
d
 P
o
w
e
r
θ = 0˚
θ = 30˚
θ = 50˚
 
0
0.2
0.4
0.6
0.8
1
2 4 6 8 10 12
Frequency (GHz)
R
e
fl
e
c
te
d
 P
o
w
e
r
θ = 0˚
θ = 30˚
θ = 50˚
 
Fig. 3.3 – The transmitted and reflected powers as a function 
of  frequency for three values of the angle of incidence   0 , 
  30 ,   50  
 
Fig. 3.4 represents the transmitted and reflected 
powers versus the ferrite thickness for three values of 
the applied magnetic fields, 30  angle of incidence and  
 
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 12 14 16 18
Thickness of Ferrite (mm)
T
ra
n
s
m
it
te
d
 P
o
w
e
r
B = 500 Oe
B = 750 Oe
B = 1000 Oe
 
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 2 4 6 8 10 12 14 16 18
Thickness of Ferrite (mm)
R
e
fl
e
c
te
d
 P
o
w
e
r
B = 500 Oe
B = 750 Oe
B = 1000 Oe
 
Fig. 3.4 – The transmitted and reflected powers versus ferrite 
thickness for three values of applied magnetic fields 
B  500 Oe, B  750 Oe, and B  1000 Oe 
a 
b 
a 
b 
b 
a 
 MUIN F. UBEID, ET AL. J. NANO- ELECTRON. PHYS. 4, 01009 (2012) 
 
 
01009-4 
8.6 GHz frequency of the incident waves. The slab 
thickness is changed from zero mm to /2 (17.44 mm). 
It can be observed that the reflected power shows oscil-
latory behavior with thickness for all values of the ap-
plied magnetic fields. The transmitted power shows the 
same behavior but slightly decreasing with thickness 
for B  500 Oe and 750 Oe and considerably decreasing 
with thickness for B  1000 Oe. 
 
4. CONCLUSIONS 
 
Reflection and transmission of electromagnetic waves 
by a magnetized ferrite-wire structure inserted in vac-
uum are analyzed numerically. The total reflection of 
the waves by the structure is obtained due to the dif-
ference in signs between  and  of the structure. The 
transmission of radiations through the structure is 
realized when  and  of the structure are both negative 
or both positive. The frequency band of reflection and 
transmission of the waves by the structure can be tuned 
by changing the applied magnetic field due to the tena-
bility of  and . Thus we can say that the variation of 
the applied magnetic field changes the reflection and 
transmission of the waves by the structure. Numerical 
examples are already presented to illustrate the paper 
idea and to prove the validity of the obtained results. 
Moreover the conservation of energy is satisfied 
throughout the performed computations for all exam-
ples. The discussed problem is useful for applications 
which require controlling of reflected and transmitted 
powers like antenna radome, microwave, millimeter 
wave and optical devices. 
 
 
 
REFERENCES 
 
1. V.G. Veselago, Sov. Phys. Usp. 10, 509 (1968). 
2. J.B. Pendry, A.J. Holden, W.J. Sewart, I. Youngs, Phys. 
Rev. Lett. 76, 4773 (1996). 
3. J.B. Pendry A.J. Holden, D.J. Robbins, W.J. Stewart, 
IEEE T. Microw. Theory 47, 2075 (1999). 
4. G. Dewar, Proc. SPIE 5218, 140 (2003). 
5. G. Dewar, J. Appl. Phys. 97, 10Q101 (2005). 
6. B.X. Cai, M.X. Zhou, K.H. Hu, Chin. Phys. Lett. 23, 348 
(2005). 
7. J. Gong, Q. Chu, Metamaterials, 2008 International Work-
shop on, 274 (2008).  
8. Y. He, P. He, V. G. Harris, C. Vittoria, IEEE Trans. Magn. 
42, 2852 (2006). 
9. M. Augustine, S. Mathew, V. Mathew, Int. J. Infrared 
Milli. Waves 29, 534 (2008). 
 
10. H. Zhao, J. Hou, B. Li, L. Kang, Appl. Phys. Lett. 91, 
131107 (2007). 
11. R.X. Wu, J. Appl. Phys. 97, 076105 (2005). 
12. F.J. Rachford, D.N. Armstead, V. G. Harris, C. Vittoria, 
Phys. Rev. Lett. 99, 057202 (2007). 
13. H.Y. Li, Z.S. Wu, Acta Physica Sinica 57, 833 (2008). 
14. C. Sabah, S. Uckun, Opto-Elc. Rev. 15, 133 (2007). 
15. D.D. Stancil, A. Prabhakar, Spin Waves (New York: 
Springer, 2009). 
16. R.A. Shelby, Thesis (PhD.), University of Calefornia, San 
Diego, Microwave Experiments with Left-Handed Materi-
als (Bell and Howell Information and Learning Company 
2001). 
17. Y. Huang, G. Wen, K. Xie, Applied Superconductivity and 
Electromagnetic Devices, 2009. ASEMD 2009. Internation-
al Conference on, 119 (2009).  
 
 
 


English

Numerical Study of Negative-Refractive Index Ferrite Waveguide

Consider a magnetized ferrite-wire waveguide structure situated between two half
free spaces. Ferrites to provide negative permeability and wire arrays to provide negative
permittivity. The structure form left-handed material (LHM) with negative refractive
index. The transmission of electromagnetic waves through the structure is investigated
theoretically. Maxwell's equations are used to determine the electric and magnetic
fields of the incident waves at each layer. Snell's law is applied and the boundary conditions
are imposed at each layer interface to calculate the reflected and transmitted powers
of the structure. Numerical results are illustrated to show the effect of frequency,
applied magnetic fields, angle of incidence and LHM thickness on the mentioned powers.
The analyzed results show that the transmission is very good when the permeability
and permittivity of the structure are both simultaneously negative. The frequency band
corresponding to this transmission can be tuned by changing the applied magnetic
fields. The obtained results are in agreement with the law of conservation of energy.
When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/2057

Muin F. Ubed

Mohammed M. Shabat

Mohammed O. Sid-Ahmed

Electronic Sumy State University Institutional Repository

Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                   47  NUMERICAL STUDY OF  NEGATIVE-REFRACTIVE  INDEX FERRITE WAVEGUIDE Muin F. Ubeid1, Mohammed M. Shabat1**, Mohammed O. Sid-Ahmed2   1 Department of Physics, Faculty of Science, Islamic University of Gaza, P.O. 108, Gaza, Gaza Strip, Palestinian  Authority 2 Department of Physics, Faculty of Science, Sudan University of Science and Technology, The Republic of The Sudan  ABSTRACT Consider a magnetized ferrite-wire waveguide structure situated between two half free spaces. Ferrites to provide negative permeability and wire arrays to provide nega-tive permittivity. The structure form left-handed material (LHM) with negative refrac-tive index. The transmission of electromagnetic waves through the structure is investi-gated theoretically. Maxwell's equations are used to determine the electric and magnetic fields of the incident waves at each layer. Snell's law is applied and the boundary condi-tions are imposed at each layer interface to calculate the reflected and transmitted pow-ers of the structure. Numerical results are illustrated to show the effect of frequency, applied magnetic fields, angle of incidence and LHM thickness on the mentioned pow-ers. The analyzed results show that the transmission is very good when the permeability and permittivity of the structure are both simultaneously negative. The frequency band corresponding to this transmission can be tuned by changing the applied magnetic fields. The obtained results are in agreement with the law of conservation of energy.   Key words: Applied magnetic fields, Electromagnetic waves, ferromagnetic ma-terial, frequency, Reflected power, transmitted power.  INTRODUCTION Metamaterials (sometimes termed left-handed materials (LHMs)) are ma-terials whose permittivity H  and permeability P  are both negative and conse-quently have negative index of refraction. These materials are artificial and theoretically discussed first by Veselago [1] over 40 years ago. The first reali-zation of such materials, consisting of split-ring resenators (SRRs) and continu-ous wires, was first introduced by Pendry [2, 3]. Regular materials are materials whose H  and P  are both positive and termed right handed materials (RHMs). Magnetized ferrite is an additional alternative to SRR to provide negative permeability. Consequently, a LHM can be achieved by inserting a periodic continuous wires into the ferrite material [4-7]. İ and ȝ of this LHM have tuna-                                               * e-mail: shabatm@gmail.com, tel.: (+970)82860700 fax: (+970)82860800 48                   Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I             ble properties by changing the applied bias magnetic fields and both exhibit negative values at certain frequency band. Y. He et al [8] have studied the role of ferrites in negative index metamaterials. M. Augustine et al [9] have formu-lated a theoretical analysis of ferrite-superconductor layered structures. H. Zhao et al [10] have studied a magnetotunable left-handed material consisting of yttrium iron garnet slab and metallic wires. R. X. Wu. [11] has shown the effect of negative refraction index in periodic metal-ferrite film composite. F. J. Rach-ford et al [12] have performed simulations of ferrite-dielectric-wire composite negative index materials. Q. X. Chu et al [13] have shown a novel left-handed metamaterial with wire array in a ferrite-filled rectangular waveguide.  In this paper we consider a magnetized ferrite-wire structure inserted in vacuum. A plane polarized wave is obliquely incident on it. İ and ȝ of the structure can easily be tuned by changing the applied magnetic fields and nega-tive refractive index of it is achieved at a certain frequency band to make the structure LHM. Maxwell's equations are used to determine the electric and magnetic fields in each region. Then , Snell's law is applied and the boundary conditions of the fields are imposed at each interface to obtain a number of equations with unknown parameters. The equations are solved  for the un-known parameters to calculate the reflection and transmission coefficients. These coefficients are used to determine the reflected and transmitted powers of the structure. The effect of many parameters like frequency, applied magnetic fields and angle of incidence etc. on the mentioned powers is studied in detail. It is demonstrated that, if both İ and ȝ of the ferrite-wire structure are both positive and both negative, then the electromagnetic waves can propagate through it. On the other hand if either İ or ȝ of the structure is negative, the incident radiation will be reflected and no transmission of the waves will be through the structure. Moreover the frequency band of zero or nonzero trans-mission of the waves can be tuned by changing the applied magnetic fields due to the tenability of İ and ȝ of the structure. To check the validity of the per-formed computations the law of conservation of energy given by [14 , 15 ] is satisfied for all examples.  THEORY Consider a rectangular ferrite-wire waveguide structure located between two half free spaces and a dc applied magnetic field acts on it along the X axis [4, 5]. A plane polarized wave is incident on the plane y = 0 at angle T  relative to the normal to the boundary (fig. 1). The effective permeability of the ferrite is given by [4, 5]:                                   KKP22 . f                                               (1)  Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                   49  Where 22)()(1ZDZZDZZZK iioom , 22)( ZDZZZZ .iom  , Bo JZ  , Mm JZ   Ȗ is the gyromagnetic ratio, B is the intensity of the applied magnetic field, M is the saturation magnetization, Ȧ is the angular frequency and i = 1 . The effective permittivity of the structure is considered to be [4, 5]: )]413.18.3)(1057.1([ 6 foeffoeffrf i PPZVZHVHHu        (2) Where oH  and oP  are the permittivity and permeability of free space re-spectively, rH  is the relative permittivity of ferrite material and effV is the effective conductivity of wire arrays.     Fig. 1 – Oblique incidence of electromagnetic wave on a ferrite-wire structure embedded in vacuum  The transverse components of the electric and magnetic fields in each re-gion of Fig.1 are [9, 16]: Region 1:        )(1 tzkiyikyikx ozoyoy eBeAeE Z                               (3)                      > @ )(1 )(1 tzkiyikoyyikoyozozoyoy eeBkeAkH ZZP       (4) Region 2:    tzkiyikyikx fzfyfy eDeCeE Z 2                           (5) 50                   Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I              > @  tzkiyikbyikcz fzfyfy eeDkeCkH ZZ  12                  (6) With: ffyfzckkk PPQ , ffyfzbkkk PPQ , .. i22KPQ                (7) Region 3:  tzkiyikxozoy eFeE Z 3                                        (8)    tzkiyikoyozozoy eeFkH ZZP 13                  (9) Where A, B, C, D, F are the amplitudes of the forward and backward trav-eling waves. ckoZ  is the free space wave vector, cnk ffZ  is the wave vector inside the  slab and oofffn PHPH  is the refractive index of it.  Tsinofzoz kkk   (Snell's Law)                      (10)  TZ 2sin1 ckoy , TZ 22 sin ffy nck                  (11)  Matching the boundary conditions for the transverse field components at each layer interface, that is at y=0, xx EE 21   and zz HH 21  , at y=a xx EE 32   and zz HH 32  . This yields the following equations [16]: A+B=C+D                                                         (12)   bcooy DkCkBAk P                                   (13) aikaikaik oyfyfy FeDeCe                                     (14) aikooyaikbaikcoyfyfy FekeDkeCk P                    (15) Letting A=1 and solving these four equations for the unknown parameters enables us to calculate the reflection and transmission coefficients B and F [16]. The reflected and transmitted powers are defined as [16]:                                  *BBR                                                      (16)                                  *FFT                                                      (17) Where *B _ and *F _ are the complex conjugate of B and F respectively. Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                   51  NUMERICAL RESULTS  In this section the reflected and transmitted powers of the structure are calculated numerically as a function of frequency, angle of incidence, applied magnetic fields and thickness of ferrite. In our method we have used the pa-rameters as in [17]: M = 4398 G,  J 2.8 MHz/Oe, 006. D ,  İr = 4. Three values of B are selected to be B = 500 Oe , 750 Oe, and 1000 Oe. The central frequency is selected to be 8.6 GHz. This frequency is chosen such that İf and ȝf are both simultaneously negative for all values of the applied magnetic fields. The slab thickness is assumed to be one half-wavelength long at the central frequency. ȝf Ȧ)  with  real  part  Re  (ȝf)  and İf Ȧ)  with  real  Re  (İf) are plotted as a function of frequency Ȧ in fig. 2 for three values of the applied magnetic fields ( B = 500 Oe, B = 750 Oe, B = 1000 Oe). As shown from the figure the fre-quency ranges in which Re (ȝf) and Re (İf)  are negative can be changed with the applied magnetic field [8]. In the case of B = 500 Oe, ȝf and  İf are both negative in the frequency range of (6.4 GHz – 10.8 GHz) and consequently the ferrite-wire structure exhibits negative refractive index (i. e. LHM) in that range. For B = 750 Oe and B = 1000 Oe the ferrite-wire structure exhibits LHM in. the frequency ranges of (7.2 GHz – 11 GHz) and (8.2 GHz – 11.2 GHz) respectively.    Fig.  2  – Calculated effective permeability and permittivity under different applied magnetic fields B = 500 Oe, B = 750 Oe and B = 1000 Oe  The transmitted and reflected powers as a function of frequency are calcu-lated in fig. 3.  for  ș =  30Û and  a  =  Ȝ/2.  The  frequency is  changed between 2  GHz and 12 GHz, because the simultaneously negative values of Re (ȝf) and Re İf)  under the three values of B can be realized in this range. In this example it -60-50-40-30-20-1001020304050602 3 4 5 6 7 8 9 10 11 12Frequency (GHz)Real permeability and permittivityRe (ȝf )     B = 500 OeB = 750 OeB = 1000 OeRe (İf) B = 500 OeB = 750 OeB = 1000 Oe52                   Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I             can  be  seen  that,  if  both  ȝf and  İf are both positive and both negative, the transmission is very good and the electromagnetic wave can propagate through the ferrite-wire structure. This is because the propagation constant of the waves in the structure are real  (fff ck PHZ222  ). If either ȝf or İf is negative, the prop-agation constant is imaginary and then the incident radiation will be reflected and no transmission of the waves will be through the structure. The symmetry of nonzero transmission shifts in frequency to the right side when the applied field increased. This is due to the tunability of the LHM which arises from the variation of the applied magnetic fields.  (a)  (b) Fig. 3 – The transmitted and reflected powers against frequency when the ap-plied magnetic field changes B = 500 Oe, B = 750 Oe and B = 1000 Oe 00.20.40.60.812 4 6 8 10 12Frequency (GHz)Transmitted PowerB = 500 OeB = 750 OeB = 1000 Oe00.20.40.60.810 2 4 6 8 10 12Frequency (GHz)Reflected PowerB = 500 OeB = 750 OeB = 1000 OeNanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                   53  Figure 4 shows the transmitted and reflected powers as a function of fre-quency when the incidence angle changes (ș = 0Û, ș = 30Û, ș = 50Û) for B = 500 Oe. The frequency range and thickness of ferrite is the same as in the last ex-ample. It can be seen that the transmitted power is decreasing while the reflect-ed power increasing with the angle of incidence. Moreover they show oscillato-ry behavior for ș = 30Û and 50Û while they show nearly no ripples for ș = 0Û.   (a)  (b) Fig. 4 – The transmitted and reflected powers as a function of  frequency  for three values of the angle of incidence ș = 0Û, ș = 30Û, ș = 50Û. 00.20.40.60.812 4 6 8 10 12Frequency (GHz)Transmitted Powerș = 0Ûș = 30Ûș = 50Û00.20.40.60.812 4 6 8 10 12Frequency (GHz)Reflected Powerș = 0Ûș = 30Ûș = 50Û54                   Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I             Figure 5 represents the transmitted and reflected powers versus the ferrite thickness for three values of the applied magnetic fields, 30Û angle of incidence and 8.6 GHz frequency of the incident waves. The slab thickness is changed from zero mm to Ȝ/2 (17.44 mm). It  can be observed that the reflected power shows oscillatory behavior with thickness for all values of the applied magnetic fields. The transmitted power shows the same behavior but slightly decreasing with thickness for B = 500 Oe and 750 Oe and considerably decreasing with thickness for B = 1000 Oe.   (a)  (b) Fig. 5 – The transmitted and reflected powers versus ferrite thickness for three values of  applied magnetic fields B = 500 Oe, B = 750 Oe and B = 1000 Oe  0.40.50.60.70.80.910 2 4 6 8 10 12 14 16 18Thickness of Ferrite (mm)Transmitted PowerB = 500 OeB = 750 OeB = 1000 Oe00.050.10.150.20.250.30.350 2 4 6 8 10 12 14 16 18Thickness of Ferrite (mm)Reflected PowerB = 500 OeB = 750 OeB = 1000 OeNanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                   55  CONCLUSIONS Reflection and transmission of electromagnetic waves by a magnetized ferrite-wire structure inserted in vacuum are analyzed numerically. Total reflec-tion of the waves by the structure is obtained due to the difference in signs between ȝ and İ of the structure. Transmission of radiations through the struc-ture is realized when ȝ and İ of the structure are both negative or both positive. The frequency band of reflection and transmission of the waves by the structure can be tuned by changing the applied magnetic field due to the tunability of ȝ and İ. Thus we can say that the variation of the applied magnetic field changes the reflection and transmission of the waves by the structure. Numerical exam-ples are already presented to illustrate the paper idea and to prove the validity of the obtained results. Moreover the conservation of energy is satisfied throughout the performed computations for all examples. The discussed prob-lem is useful for applications which require controlling of reflected and trans-mitted powers like antenna radome, microwave, millimeter wave and optical devices.  REFERENCES [1]. V.G. Veselago, Soviet Phys Uspekhi, 1986 Vol. 10, P. 509-514. [2]. J.B. Pendry, A.J. Holden, W.J. Sewart, I. Youngs, Phys. Rev. Lett., 1996, Vol. 76, P. 4773-4776. [3]. J.B. Pendry A.J. Holden, D.J. Robbins, W.J. Stewart, IEEE  Transaction on Micro-wave Theory and Techniques, 1999, Vol. 47, P. 2075-2084. [4]. G. Dewar, Proceeding of SPIE, 2003, 5218:140-144. . [5]. G. Dewar, J. Appl. Phys., 2005, Vol. 97, 10Q101: 1-3.  [6]. B. X. Cai, M.X. Zhou and K.H. Hu, Chin. Phys. Lett., 2005, Vol.  23, P. 348-351. [7]. J. Gong and Q. Chu, Proceeding of Digital Object Identifier, 2008,  PP. 274-277.  [8]. Y. He, P. He, V. G. Harris and C. Vittoria, IEEE Trans.  Magnetized, 2006, Vol. 42, P. 2852.  [9]. M. Augustine, S. Mathew and V. Mathew, Int. J. Infrared Milli  Waves, 2008, Vol. 29, P. 534-544. [10]. H. Zhao, J. hou, B. Li and L. Kang, Appl. Phys.Lett., 2007, Vol. 91, P. 131107(1-2).  [11]. R. X. Wu., Journal of Applied Physics, 2005, Vol. 97, No. 7,  P. 076105.  [12]. F.  J.  Rachford,  D.  N.  Armstead,  V.  G.  Harris  and  C.  Vittoria,  Phys.  Rev.  Lett.,  2007, Vol. 99(057202), P. 1-4. [13]. Q. X. Chu and J. Q. Gong, ACTA PHYSICA SINICA, 2008,  Vol. 57, P. 2925-2927. [14]. C. Sabah and S. Uckun, Opto-Elc. Rev., 2007, Vol. 15, No. 3, P. 133-143.  [15]. D. D. Stancil and A Prabhakar, Spin Waves, New York:  Springer, 2009.  [16]. R.A. Shelby, Thesis (PhD.), University of California, San Diego,  Bell and Howell Information and Learning Company 2001. [17]. Y Huang, G Wen and K Xie, Proceeding of ASEMD, 2009, P.  119-122.    

Numerical study of negative-refractive index ferrite waveguide

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ Vol. 4 No 1, 01009(4pp) (2012) Том 4 № 1, 01009(4cc) (2012)   2077-6772/2012/4(1)01009(4) 01009-1  2012 Sumy State University Numerical Study of Negative-Refractive Index Ferrite Waveguide  Muin F. Ubeid1,*, Mohammed M. Shabat1, Mohammed O. Sid-Ahmed2  1 Department of Physics, Faculty of Science, Islamic University of Gaza, P.O. Box 108, Gaza,  Gaza Strip, Palestinian Authority 2 Department of Physics, Faculty of Science, Sudan University of Science and Technology,  The Republic of the Sudan  (Received 22 October 2011; revised manuscript received 26 February 2012; published online 14 March 2012)  Consider a magnetized ferrite-wire waveguide structure situated between two half free spaces. Ferrites to provide negative permeability and wire array to provide negative permittivity. The structure form left-handed material (LHM) with negative refractive index. The transmission of electromagnetic waves through the structure is investigated theoretically. Maxwell's equations are used to determine the electric and magnetic fields of the incident waves at each layer. Snell's law is applied and the boundary conditions are imposed at each layer interface to calculate the reflected and transmitted powers of the structure. Nu-merical results are illustrated to show the effect of frequency, applied magnetic fields, angle of incidence and LHM thickness on the mentioned powers. The analyzed results show that the transmission is very good when the permeability and permittivity of the structure are both simultaneously negative. The fre-quency band corresponding to this transmission can be tuned by changing the applied magnetic fields. The obtained results are in agreement with the law of conservation of energy.  Keywords: Applied magnetic fields, Electromagnetic waves, Ferromagnetic material, Frequency, Reflect-ed power, Transmitted power.   PACS numbers: 42.25Bs, 42.25.Dd, 42.25.Gy   ______________ * mubeid@mail.iugaza.edu 1. INTRODUCTION  Metamaterials (sometimes termed left-handed ma-terials (LHMs)) are materials whose both permittivity  and permeability  are negative and consequently have negative index of refraction. These materials are artifi-cial and theoretically discussed first by Veselago [1] over 40 years ago. The first realization of such materi-als, consisting of split-ring resonators (SRRs) and con-tinuous wires, was first introduced by Pendry [2, 3]. Regular materials are materials with positive  and  and termed right handed materials (RHMs). Magnetized ferrite is an additional alternative to SRR to provide negative permeability. Consequently, a LHM can be achieved by inserting a periodic continu-ous wires into the ferrite material [4-7].Both  and  of this LHM have tunable properties by changing the ap-plied bias magnetic fields and both exhibit negative values at certain frequency band. Y. He et al. [8] have studied the role of ferrites in negative index metamate-rials. M. Augustine et al. [9] have formulated a theoret-ical analysis of ferrite-superconductor layered struc-tures. H. Zhao et al. [10] have studied a magnetotuna-ble left-handed material consisting of yttrium iron gar-net slab and metallic wires. R.X. Wu [11] has shown the effect of negative refraction index in periodic metal-ferrite film composite. F.J. Rachford et al. [12] have performed simulations of ferrite-dielectric-wire compo-site negative index materials. Q.X. Chu et al. [13] have shown a novel left-handed metamaterial with wire ar-ray in a ferrite-filled rectangular waveguide.  In this paper we consider a magnetized ferrite-wire structure inserted in vacuum. A plane polarized wave is obliquely incident on it.  and  of the structure can easily be tuned by changing the applied magnetic fields and negative refractive index of it is achieved at a cer-tain frequency band to make the structure as LHM. Maxwell's equations are used to determine the electric and magnetic fields in each region. Then, Snell's law is applied and the boundary conditions of the fields are imposed at each interface to obtain a number of equa-tions with unknown parameters. The equations are solved for the unknown parameters to calculate the reflection and transmission coefficients. These coeffi-cients are used to determine the reflected and trans-mitted powers of the structure. The effect of many pa-rameters like frequency, applied magnetic fields and angle of incidence etc. on the mentioned powers are studied in detail. It is demonstrated that, if both ε and μ of the ferrite-wire structure are both positive or both negative, then the electromagnetic waves can propa-gate through it. On the other hand if either  or  of the structure is negative, the incident radiation will be reflected and no transmission of the waves will be through the structure. Moreover the frequency band of zero or nonzero transmission of the waves can be tuned by changing the applied magnetic fields due to the ten-ability of ε and μ of the structure. To check the validity of the performed computations the law of conservation of energy given by [14, 15 ] is satisfied for all examples.  2. THEORY  Consider a rectangular ferrite-wire waveguide structure located between two half free spaces and a dc applied magnetic field acts on it along the X axis [4, 5]. A plane polarized wave is incident on the plane y = 0 at angle  relative to the normal to the boundary (Fig.1). The effective permeability the ferrite is given by [4, 5]:   2 2f , (2.1)  MUIN F. UBEID, ET AL. J. NANO- ELECTRON. PHYS. 4, 01009 (2012)   01009-2 where 02 20( )1( )m ii, 2 20( )mi,    0 B , m M ,  is the gyromagnetic ratio, B is the intensity of the applied magnetic field, M is the saturation magnetization, ω is the angular frequency and i = 1 . The effective permittivity of the structure is considered to be [4, 5]:   60 0[ (1.57 10 )(3.18 .413 )]efff reff fi (2.2)  Where 0 and 0 are the permittivity and permeabil-ity of free space respectively, r is the relative permit-tivity of ferrite material and eff is the effective conduc-tivity of wire arrays.     Fig. 2.1 – Oblique incidence of electromagnetic wave on a ferrite-wire structure embedded in vacuum  The transverse components of the electric and mag-netic fields in each region of Fig. 2.1 are [9, 16]: Region 1:   ( )1 ( )oy oy ozik y ik y i k z txE Ae Be e  (2.3)  ( )101( )oy oy ozik y ik y i k z tz oy oyH Ak e Bk e e  (2.4)  Region 2:   ( )2 ( )fy fy fzik y ik y i k z txE Ce De e  (2.5)   ( )21( )fy fy fzik y ik y i k z tz c bH Ck e Dk e e  (2.6)  With:  fz fycfk kk , fz fybfk kk , 2 2i (2.7)  Region 3:   ( )3 oy ozik y i k z txE Fe e   (2.8)  ( )301( )oy ozik y i k z tz oyH Fk e e  (2.9)  Where A, B, C, D, F are the amplitudes of the for-ward and backward traveling waves. k0  /c is the free space wave vector, kf  nfω/c is the wave vector inside the  slab and 0 0f f fn  is the refractive index of it.  sinoz fz ok k k  (Snell's Law) (2.10)   21 sinoykc, 2 2sinfy fk nc (2.11)  Matching the boundary conditions for the trans-verse field components at each layer interface, that is at y  0, 1 2x xE E  and 1 2z zH H , at y  a 2 3x xE E  and 2 3z zH H . This yields the following equations [16]:   A + B  C + D (2.12)   0( )oyc bkA B Ck Dk   (2.13)   fy fy oyik a ik a ik aCe De Fe  (2.14)    0fy fy oyoyik a ik a ik ac bkCk e Dk e Fe  (2.15)  Letting A  1 and solving these four equations for the unknown parameters enables us to calculate the reflec-tion and transmission coefficients B and F [16]. The re-flected and transmitted powers are defined as [16]:   R  BB* (2.16)   T  FF* (2.17)  Where B* and F* are the complex conjugate of B and F respectively.  3. NUMERICAL RESULTS AND DISCUSSION   In this section the reflected and transmitted powers of the structure are calculated numerically as a func-tion of frequency, angle of incidence, applied magnetic fields and thickness of ferrite. In our method we have used the parameters as in [17]: M  4398 G,   2.8 MHz/Oe,   0.006, r  4. Three values of B are selected to be B  500 Oe, 750 Oe, and 1000 Oe. The central frequency is selected to be 8.6 GHz. This fre-quency is chosen such that f and f are both simulta-neously negative for all values of the applied magnetic fields. The slab thickness is assumed to be one half-wavelength long at the central frequency. f ( ) with real part Re ( f) and f( ) with real Re( f) are plotted as a function of frequency  in Fig. 2.2 for three values of the applied magnetic fields ( B  500 Oe, B  750 Oe, B  1000 Oe). As shown from the figure the frequency range in which Re( f) and Re( f) are negative can be changed with the applied magnetic field [8]. In the case of B  500 Oe, f and f are both negative in the frequency range of (6.4-10.8 GHz) and consequently the ferrite-wire structure exhibits negative refractive index (i.e. LHM) in that range. For B  750 Oe and B  1000 Oe the ferrite-wire structure exhibits LHM in the frequency ranges of (7.2-11 GHz) and (8.2-11.2 GHz) respectively. The transmitted and reflected powers as a function of frequency are calculated in Fig. 3.2. for   30  and a  /2. The frequency is changed between 2 GHz and 12 GHz, because the simultaneously negative values of Re( f) and Re( f) under the three values of B can be  NUMERICAL STUDY OF NEGATIVE-REFRACTIVE INDEX… J. NANO- ELECTRON. PHYS. 4, 01009 (2012)   01009-3 -60-50-40-30-20-1001020304050602 3 4 5 6 7 8 9 10 11 12Frequency (GHz)Real permeability and permittivity Re (μf)     B = 500 OeB = 750 OeB = 1000 OeRe (εf) B = 500 OeB = 750 OeB = 1000 Oe Fig. 3.1 – Calculated effective permeability and permittivity under different applied magnetic fields B  500 Oe, B  750 Oe nd B  1000 Oe  realized in this range. In this example it can be seen that, if both f and f are both positive or both negative, the transmission is very good and the electromagnetic wave can propagate through the ferrite-wire structure. This is because the propagation constant of the waves in the structure is real ( 2 2 2f f fk c ). If either f or f is negative, the propagation constant is imaginary and then the incident radiation will be reflected and no transmission of the waves will be through the structure. The symmetry of nonzero transmission shifts in frequen-cy to the right side when the applied field increased. This is due to the tunability of the LHM which arises from the variation of the applied magnetic fields. 00.20.40.60.812 4 6 8 10 12Frequency (GHz)Transmitted PowerB = 500 OeB = 750 OeB = 1000 Oe 00.20.40.60.810 2 4 6 8 10 12Frequency (GHz)Reflected PowerB = 500 OeB = 750 OeB = 1000 Oe Fig. 3.2 – The transmitted and reflected powers against fre-quency when the applied magnetic field changes B  500 Oe, B  750 Oe, and B  1000 Oe  Fig. 3.3 shows the transmitted and reflected powers as a function of frequency when the incidence angle changes (   0 ,   30 ,   50 ) for B  500 Oe. The frequency range and thickness of ferrite is the same as in the last example. It can be seen that the transmitted power is decreasing while the reflected power increas-ing with the angle of incidence. Moreover they show oscillatory behavior for   30  and 50  while they show nearly no ripples for   0 . 00.20.40.60.812 4 6 8 10 12Frequency (GHz)Transmitted Powerθ = 0˚θ = 30˚θ = 50˚ 00.20.40.60.812 4 6 8 10 12Frequency (GHz)Reflected Powerθ = 0˚θ = 30˚θ = 50˚ Fig. 3.3 – The transmitted and reflected powers as a function of  frequency for three values of the angle of incidence   0 ,   30 ,   50   Fig. 3.4 represents the transmitted and reflected powers versus the ferrite thickness for three values of the applied magnetic fields, 30  angle of incidence and   0.40.50.60.70.80.910 2 4 6 8 10 12 14 16 18Thickness of Ferrite (mm)Transmitted PowerB = 500 OeB = 750 OeB = 1000 Oe 00.050.10.150.20.250.30.350 2 4 6 8 10 12 14 16 18Thickness of Ferrite (mm)Reflected PowerB = 500 OeB = 750 OeB = 1000 Oe Fig. 3.4 – The transmitted and reflected powers versus ferrite thickness for three values of applied magnetic fields B  500 Oe, B  750 Oe, and B  1000 Oe a b a b b a  MUIN F. UBEID, ET AL. J. NANO- ELECTRON. PHYS. 4, 01009 (2012)   01009-4 8.6 GHz frequency of the incident waves. The slab thickness is changed from zero mm to /2 (17.44 mm). It can be observed that the reflected power shows oscil-latory behavior with thickness for all values of the ap-plied magnetic fields. The transmitted power shows the same behavior but slightly decreasing with thickness for B  500 Oe and 750 Oe and considerably decreasing with thickness for B  1000 Oe.  4. CONCLUSIONS  Reflection and transmission of electromagnetic waves by a magnetized ferrite-wire structure inserted in vac-uum are analyzed numerically. The total reflection of the waves by the structure is obtained due to the dif-ference in signs between  and  of the structure. The transmission of radiations through the structure is realized when  and  of the structure are both negative or both positive. The frequency band of reflection and transmission of the waves by the structure can be tuned by changing the applied magnetic field due to the tena-bility of  and . Thus we can say that the variation of the applied magnetic field changes the reflection and transmission of the waves by the structure. Numerical examples are already presented to illustrate the paper idea and to prove the validity of the obtained results. Moreover the conservation of energy is satisfied throughout the performed computations for all exam-ples. The discussed problem is useful for applications which require controlling of reflected and transmitted powers like antenna radome, microwave, millimeter wave and optical devices.    REFERENCES  1. V.G. Veselago, Sov. Phys. Usp. 10, 509 (1968). 2. J.B. Pendry, A.J. Holden, W.J. Sewart, I. Youngs, Phys. Rev. Lett. 76, 4773 (1996). 3. J.B. Pendry A.J. Holden, D.J. Robbins, W.J. Stewart, IEEE T. Microw. Theory 47, 2075 (1999). 4. G. Dewar, Proc. SPIE 5218, 140 (2003). 5. G. Dewar, J. Appl. Phys. 97, 10Q101 (2005). 6. B.X. Cai, M.X. Zhou, K.H. Hu, Chin. Phys. Lett. 23, 348 (2005). 7. J. Gong, Q. Chu, Metamaterials, 2008 International Work-shop on, 274 (2008).  8. Y. He, P. He, V. G. Harris, C. Vittoria, IEEE Trans. Magn. 42, 2852 (2006). 9. M. Augustine, S. Mathew, V. Mathew, Int. J. Infrared Milli. Waves 29, 534 (2008).  10. H. Zhao, J. Hou, B. Li, L. Kang, Appl. Phys. Lett. 91, 131107 (2007). 11. R.X. Wu, J. Appl. Phys. 97, 076105 (2005). 12. F.J. Rachford, D.N. Armstead, V. G. Harris, C. Vittoria, Phys. Rev. Lett. 99, 057202 (2007). 13. H.Y. Li, Z.S. Wu, Acta Physica Sinica 57, 833 (2008). 14. C. Sabah, S. Uckun, Opto-Elc. Rev. 15, 133 (2007). 15. D.D. Stancil, A. Prabhakar, Spin Waves (New York: Springer, 2009). 16. R.A. Shelby, Thesis (PhD.), University of Calefornia, San Diego, Microwave Experiments with Left-Handed Materi-als (Bell and Howell Information and Learning Company 2001). 17. Y. Huang, G. Wen, K. Xie, Applied Superconductivity and Electromagnetic Devices, 2009. ASEMD 2009. Internation-al Conference on, 119 (2009).     

Consider a magnetized ferrite-wire waveguide structure situated between two half free spaces. Ferrites to provide negative permeability and wire arrays to provide negative permittivity. The structure form left-handed material (LHM) with negative refractive index. The transmission of electromagnetic waves through the structure is investigated theoretically. Maxwell's equations are used to determine the electric and magnetic fields of the incident waves at each layer. Snell's law is applied and the boundary conditions are imposed at each layer interface to calculate the reflected and transmitted powers of the structure. Numerical results are illustrated to show the effect of frequency, applied magnetic fields, angle of incidence and LHM thickness on the mentioned powers. The analyzed results show that the transmission is very good when the permeability and permittivity of the structure are both simultaneously negative. The frequency band corresponding to this transmission can be tuned by changing the applied magnetic fields. The obtained results are in agreement with the law of conservation of energy. When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/2057

Ubed, Muin F

Sid-Ahmed, Mohammed O

Institutional Repository of the Islamic University of Gaza

Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                   47 
 
NUMERICAL STUDY OF  NEGATIVE-REFRACTIVE  
INDEX FERRITE WAVEGUIDE 
Muin F. Ubeid1, Mohammed M. Shabat1**, Mohammed O. Sid-Ahmed2  
 
1 Department of Physics, Faculty of Science, Islamic University of Gaza, P.O. 108, Gaza, Gaza 
Strip, Palestinian  Authority 
2 Department of Physics, Faculty of Science, Sudan University of Science and Technology, The 
Republic of The Sudan 
 
ABSTRACT 
Consider a magnetized ferrite-wire waveguide structure situated between two half 
free spaces. Ferrites to provide negative permeability and wire arrays to provide nega-
tive permittivity. The structure form left-handed material (LHM) with negative refrac-
tive index. The transmission of electromagnetic waves through the structure is investi-
gated theoretically. Maxwell's equations are used to determine the electric and magnetic 
fields of the incident waves at each layer. Snell's law is applied and the boundary condi-
tions are imposed at each layer interface to calculate the reflected and transmitted pow-
ers of the structure. Numerical results are illustrated to show the effect of frequency, 
applied magnetic fields, angle of incidence and LHM thickness on the mentioned pow-
ers. The analyzed results show that the transmission is very good when the permeability 
and permittivity of the structure are both simultaneously negative. The frequency band 
corresponding to this transmission can be tuned by changing the applied magnetic 
fields. The obtained results are in agreement with the law of conservation of energy.  
 
Key words: Applied magnetic fields, Electromagnetic waves, ferromagnetic ma-
terial, frequency, Reflected power, transmitted power. 
 
INTRODUCTION 
Metamaterials (sometimes termed left-handed materials (LHMs)) are ma-
terials whose permittivity ?  and permeability ?  are both negative and conse-
quently have negative index of refraction. These materials are artificial and 
theoretically discussed first by Veselago [1] over 40 years ago. The first reali-
zation of such materials, consisting of split-ring resenators (SRRs) and continu-
ous wires, was first introduced by Pendry [2, 3]. Regular materials are materials 
whose ?  and ?  are both positive and termed right handed materials (RHMs). 
Magnetized ferrite is an additional alternative to SRR to provide negative 
permeability. Consequently, a LHM can be achieved by inserting a periodic 
continuous wires into the ferrite material [4-7]. ? and ? of this LHM have tuna-
                                               
* e-mail: shabatm@gmail.com, tel.: (+970)82860700 fax: (+970)82860800 
48                   Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I            
 
ble properties by changing the applied bias magnetic fields and both exhibit 
negative values at certain frequency band. Y. He et al [8] have studied the role 
of ferrites in negative index metamaterials. M. Augustine et al [9] have formu-
lated a theoretical analysis of ferrite-superconductor layered structures. H. Zhao 
et al [10] have studied a magnetotunable left-handed material consisting of 
yttrium iron garnet slab and metallic wires. R. X. Wu. [11] has shown the effect 
of negative refraction index in periodic metal-ferrite film composite. F. J. Rach-
ford et al [12] have performed simulations of ferrite-dielectric-wire composite 
negative index materials. Q. X. Chu et al [13] have shown a novel left-handed 
metamaterial with wire array in a ferrite-filled rectangular waveguide.  
In this paper we consider a magnetized ferrite-wire structure inserted in 
vacuum. A plane polarized wave is obliquely incident on it. ? and ? of the 
structure can easily be tuned by changing the applied magnetic fields and nega-
tive refractive index of it is achieved at a certain frequency band to make the 
structure LHM. Maxwell's equations are used to determine the electric and 
magnetic fields in each region. Then , Snell's law is applied and the boundary 
conditions of the fields are imposed at each interface to obtain a number of 
equations with unknown parameters. The equations are solved  for the un-
known parameters to calculate the reflection and transmission coefficients. 
These coefficients are used to determine the reflected and transmitted powers of 
the structure. The effect of many parameters like frequency, applied magnetic 
fields and angle of incidence etc. on the mentioned powers is studied in detail. 
It is demonstrated that, if both ? and ? of the ferrite-wire structure are both 
positive and both negative, then the electromagnetic waves can propagate 
through it. On the other hand if either ? or ? of the structure is negative, the 
incident radiation will be reflected and no transmission of the waves will be 
through the structure. Moreover the frequency band of zero or nonzero trans-
mission of the waves can be tuned by changing the applied magnetic fields due 
to the tenability of ? and ? of the structure. To check the validity of the per-
formed computations the law of conservation of energy given by [14 , 15 ] is 
satisfied for all examples.  
THEORY 
Consider a rectangular ferrite-wire waveguide structure located between 
two half free spaces and a dc applied magnetic field acts on it along the X axis 
[4, 5]. A plane polarized wave is incident on the plane y = 0 at angle ?  relative 
to the normal to the boundary (fig. 1). The effective permeability of the ferrite 
is given by [4, 5]: 
                                  
?
??
22 ???f                                               (1)  
Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                   49 
 
Where 
22)(
)(
1
????
?????
??
???
i
i
o
om , 
22)( ????
??
??
??
io
m  , Bo ?? ?
, Mm ?? ?  
? is the gyromagnetic ratio, B is the intensity of the applied magnetic field, 
M is the saturation magnetization, ? is the angular frequency and i = 1? . 
The effective permittivity of the structure is considered to be [4, 5]: 
)]413.18.3)(1057.1([ 6 foeffo
eff
rf i ??????
???
???
?? ?       (2) 
Where o?  and o?  are the permittivity and permeability of free space re-
spectively, r?  is the relative permittivity of ferrite material and eff? is the 
effective conductivity of wire arrays.  
 
 
 
Fig. 1 – Oblique incidence of electromagnetic wave on a ferrite-wire structure 
embedded in vacuum 
 
The transverse components of the electric and magnetic fields in each re-
gion of Fig.1 are [9, 16]: 
Region 1: 
     ? ? )(1 tzkiyikyikx ozoyoy eBeAeE ?????                               (3) 
                     ? ? )(1 )(1 tzkiyikoyyikoy
o
z
ozoyoy eeBkeAkH ???
?????       (4) 
Region 2: ? ? ? ?tzkiyikyikx fzfyfy eDeCeE ?????2                           (5) 
50                   Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I            
 
? ?? ? ? ?tzkiyikbyikcz fzfyfy eeDkeCkH ?? ???? 12                  (6) 
With: 
f
fyfz
c
kk
k ???
?? , 
f
fyfz
b
kk
k ???
?? , ?
???
i
22???                (7) 
Region 3: 
? ?tzkiyik
x
ozoy eFeE ???3                                        (8) 
? ? ? ?tzkiyikoy
o
z
ozoy eeFkH ???
??? 13                  (9) 
Where A, B, C, D, F are the amplitudes of the forward and backward trav-
eling waves. 
cko
??  is the free space wave vector, c
nk ff
??  is the wave 
vector inside the  slab and 
oo
ff
fn ??
???  is the refractive index of it. 
 ?sinofzoz kkk ?? (Snell's Law)                      (10) 
 ?? 2sin1??
c
koy , ?
? 22 sin?? ffy nck
                  (11) 
 
Matching the boundary conditions for the transverse field components at 
each layer interface, that is at y=0, xx EE 21 ?  and zz HH 21 ? , at y=a 
xx EE 32 ?  and zz HH 32 ? . This yields the following equations [16]: 
A+B=C+D                                                         (12) 
? ? bc
o
oy DkCkBA
k
?????
                                   (13) 
aikaikaik oyfyfy FeDeCe ?? ?                                  (14) 
aik
o
oyaik
b
aik
c
oyfyfy Fe
k
eDkeCk ????
?                   (15) 
Letting A=1 and solving these four equations for the unknown parameters 
enables us to calculate the reflection and transmission coefficients B and F [16]. 
The reflected and transmitted powers are defined as [16]: 
                                 *BBR ?                                                     (16) 
                                 *FFT ?                                                     (17) 
Where *B _ and *F _ are the complex conjugate of B and F respectively. 
Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                   51 
 
NUMERICAL RESULTS  
In this section the reflected and transmitted powers of the structure are 
calculated numerically as a function of frequency, angle of incidence, applied 
magnetic fields and thickness of ferrite. In our method we have used the pa-
rameters as in [17]: M = 4398 G, ?? 2.8 MHz/Oe, 006.?? ,  ?r = 4. Three 
values of B are selected to be B = 500 Oe , 750 Oe, and 1000 Oe. The central 
frequency is selected to be 8.6 GHz. This frequency is chosen such that ?f and 
?f are both simultaneously negative for all values of the applied magnetic 
fields. The slab thickness is assumed to be one half-wavelength long at the 
central frequency. 
?f ??)  with  real  part  Re  (?f)  and ?f ??)  with  real  Re  (?f) are plotted as a 
function of frequency ? in fig. 2 for three values of the applied magnetic fields 
( B = 500 Oe, B = 750 Oe, B = 1000 Oe). As shown from the figure the fre-
quency ranges in which Re (?f) and Re (?f)  are negative can be changed with 
the applied magnetic field [8]. In the case of B = 500 Oe, ?f and  ?f are both 
negative in the frequency range of (6.4 GHz – 10.8 GHz) and consequently the 
ferrite-wire structure exhibits negative refractive index (i. e. LHM) in that 
range. For B = 750 Oe and B = 1000 Oe the ferrite-wire structure exhibits LHM 
in. the frequency ranges of (7.2 GHz – 11 GHz) and (8.2 GHz – 11.2 GHz) 
respectively.   
 
Fig.  2  – Calculated effective permeability and permittivity under different applied 
magnetic fields B = 500 Oe, B = 750 Oe and B = 1000 Oe 
 
The transmitted and reflected powers as a function of frequency are calcu-
lated in fig. 3.  for  ? =  30? and  a  =  ?/2.  The  frequency is  changed between 2  
GHz and 12 GHz, because the simultaneously negative values of Re (?f) and Re 
??f)  under the three values of B can be realized in this range. In this example it 
-60
-50
-40
-30
-20
-10
0
10
20
30
40
50
60
2 3 4 5 6 7 8 9 10 11 12
Frequency (GHz)
R
ea
l p
er
m
ea
bi
lit
y 
an
d 
pe
rm
itt
iv
ity
Re (?f )     
B = 500 Oe
B = 750 Oe
B = 1000 Oe
Re (?f) 
B = 500 Oe
B = 750 Oe
B = 1000 Oe
52                   Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I            
 
can  be  seen  that,  if  both  ?f and  ?f are both positive and both negative, the 
transmission is very good and the electromagnetic wave can propagate through 
the ferrite-wire structure. This is because the propagation constant of the waves 
in the structure are real  (
fff c
k ???2
2
2 ? ). If either ?f or ?f is negative, the prop-
agation constant is imaginary and then the incident radiation will be reflected 
and no transmission of the waves will be through the structure. The symmetry 
of nonzero transmission shifts in frequency to the right side when the applied 
field increased. This is due to the tunability of the LHM which arises from the 
variation of the applied magnetic fields. 
 (a) 
 (b) 
Fig. 3 – The transmitted and reflected powers against frequency when the ap-
plied magnetic field changes B = 500 Oe, B = 750 Oe and B = 1000 Oe 
0
0.2
0.4
0.6
0.8
1
2 4 6 8 10 12
Frequency (GHz)
Tr
an
sm
itt
ed
 P
ow
er
B = 500 Oe
B = 750 Oe
B = 1000 Oe
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12
Frequency (GHz)
R
ef
le
ct
ed
 P
ow
er
B = 500 Oe
B = 750 Oe
B = 1000 Oe
Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                   53 
 
Figure 4 shows the transmitted and reflected powers as a function of fre-
quency when the incidence angle changes (? = 0?, ? = 30?, ? = 50?) for B = 500 
Oe. The frequency range and thickness of ferrite is the same as in the last ex-
ample. It can be seen that the transmitted power is decreasing while the reflect-
ed power increasing with the angle of incidence. Moreover they show oscillato-
ry behavior for ? = 30? and 50? while they show nearly no ripples for ? = 0?.   
(a) 
 (b) 
Fig. 4 – The transmitted and reflected powers as a function of  frequency  
for three values of the angle of incidence ? = 0?, ? = 30?, ? = 50?. 
0
0.2
0.4
0.6
0.8
1
2 4 6 8 10 12
Frequency (GHz)
Tr
an
sm
itt
ed
 P
ow
er
? = 0?
? = 30?
? = 50?
0
0.2
0.4
0.6
0.8
1
2 4 6 8 10 12
Frequency (GHz)
R
ef
le
ct
ed
 P
ow
er
? = 0?
? = 30?
? = 50?
54                   Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I            
 
Figure 5 represents the transmitted and reflected powers versus the ferrite 
thickness for three values of the applied magnetic fields, 30? angle of incidence 
and 8.6 GHz frequency of the incident waves. The slab thickness is changed 
from zero mm to ?/2 (17.44 mm). It  can be observed that the reflected power 
shows oscillatory behavior with thickness for all values of the applied magnetic 
fields. The transmitted power shows the same behavior but slightly decreasing 
with thickness for B = 500 Oe and 750 Oe and considerably decreasing with 
thickness for B = 1000 Oe.  
 (a) 
 (b) 
Fig. 5 – The transmitted and reflected powers versus ferrite thickness for three values of  
applied magnetic fields B = 500 Oe, B = 750 Oe and B = 1000 Oe 
 
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 12 14 16 18
Thickness of Ferrite (mm)
Tr
an
sm
itt
ed
 P
ow
er
B = 500 Oe
B = 750 Oe
B = 1000 Oe
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 2 4 6 8 10 12 14 16 18
Thickness of Ferrite (mm)
Re
fle
ct
ed
 P
ow
er
B = 500 Oe
B = 750 Oe
B = 1000 Oe
Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                   55 
 
CONCLUSIONS 
Reflection and transmission of electromagnetic waves by a magnetized 
ferrite-wire structure inserted in vacuum are analyzed numerically. Total reflec-
tion of the waves by the structure is obtained due to the difference in signs 
between ? and ? of the structure. Transmission of radiations through the struc-
ture is realized when ? and ? of the structure are both negative or both positive. 
The frequency band of reflection and transmission of the waves by the structure 
can be tuned by changing the applied magnetic field due to the tunability of ? 
and ?. Thus we can say that the variation of the applied magnetic field changes 
the reflection and transmission of the waves by the structure. Numerical exam-
ples are already presented to illustrate the paper idea and to prove the validity 
of the obtained results. Moreover the conservation of energy is satisfied 
throughout the performed computations for all examples. The discussed prob-
lem is useful for applications which require controlling of reflected and trans-
mitted powers like antenna radome, microwave, millimeter wave and optical 
devices.  
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Numerical Study of Negative-Refractive Index Ferrite Waveguide

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