Accepted 10/6/2013 Abstract: This work considers magnetostatic surface waves propagation in parallel structure composed of ferrite film bounded from below by metamaterial (MTM) placed on metal substrate and from top by air. The Maxwell equations are used to analyse propagation of slow (magnetostatic) surface electromagnetic wave in this structure. Results show that magnetostatic surface wave (MSSW) propagates at certain range of frequencies and varies as both MTM layer and ferrite layer thicknesses change. The new behaviour of the proposed waveguide in the presence of MTM in particular the appearance of the unidirectional MSSW is promising for practical applications such as isolators

Al-Sahhar, Zeyad I

El-Khozondar, Hala J.

Shabat, Mohammed M.

English

Institutional Repository of the Islamic University of Gaza

Journal of Al Azhar University-Gaza (Natural Sciences),    2013, 15 : 59-70 
http://www.alazhar.edu.ps/journal123/natural_Sciences.asp?typeno=1 
Magnetostatic Surface Waves Propagation at the 
interface between Ferrite and MTMs Parallel Plate 
Waveguide structure 
Zeyad I.Al-Sahhar1, Hala J. El-Khozondar2*1, Mohammed 
M. Shabat3 
1
Physics Department, Al-Aqsa University, Gaza, Palestine  
2
Electrical and Computer Engineering Department, Islamic 
University, P.O. Box 108, Gaza, Palestine, 
hkhozondar@iugaza.edu.ps 
3
Physics Department, Islamic University, Gaza, Palestine 
Received 11/1/2013 Accepted 10/6/2013 
Abstract:   This work considers magnetostatic surface waves propagation in 
parallel structure composed of ferrite film bounded from below by 
metamaterial (MTM) placed on metal substrate and from top by air. The 
Maxwell equations are used to analyse propagation of slow (magnetostatic) 
surface electromagnetic wave in this structure. Results show that 
magnetostatic surface wave (MSSW) propagates at certain range of 
frequencies and varies as both MTM layer and ferrite layer thicknesses 
change. The new behaviour of the proposed waveguide in the presence of 
MTM in particular the appearance of the unidirectional MSSW is promising 
for practical applications such as isolators. 
Keywords: Metamaterials, Ferrite, magnetostatic surface waves, surface 
waves. 
1. Introduction 
Metamaterials (MTMs) are media with simultaneous negative 
permittivity and permeability at a certain frequency.  In these MTMs, 
the directions of the electric field, the magnetic field, and the wave 
propagation vector obey the left-hand rule, thus MTMs are also 
known as left-handed materials materials (LHMs). Veselago (1968)  
theoretically predicted numerous unusual electromagnetic phenomena 
of these MTMs, such as a sign variation of group velocity, negative 
refraction, and perfect lensing. Attempts to achieve negative 
permittivity by using metallic wires in the GHz band were made by 
Pendry et al. (1996), which was like the plasmons of metals in optical 
frequency regions. Further, Pendry et al. (1999) suggested that an 
effective negative permeability could be attained in the GHz band 
                                                 
1 *Corresponding author. Email:hkhozondar@iugaza.edu.ps
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from a periodic arrangement of split ring resonators (SRRs). Pendry et 
al. (1996) showed that a combination of ‘‘magnetic atoms’’ and 
‘‘electric atoms’’ with negative permeability and permittivity, 
respectively, can lead to materials with a negative index of refraction. 
These materials open a new part of photonics associated with new 
concepts and potential applications (Pendry, 2000; Smith, 2004). 
Smith et al. (2000)  assembled metallic wires and SRRs to create 
simultaneously negative permittivity and permeability in the desired 
frequency range.  Kim et. al. (2006) investigated the guided dispersion 
characteristics of grounded slab waveguide structures which have 
practical applications in the area of microwave and millimeter wave 
circuits and antennas. A substantial amount of research has been 
conducted to study MTMs application in communication application 
such as isolators (El-Khozondar, H., 2008) and sensors (El-
Khozondar, R., 2008). 
Ferrites are magnetic materials characterized by anisotropic properties 
and various energetic interactions, such as dipole, exchange 
magnetoelastic and magnetooptical. The crystal structures of ferrites 
are the body-centered cubic lattice.  Their electrical properties are 
semiconductors. Their low conductivity combined with a high 
magnetic permeability made them a most valuable material for 
transformer cores (Snoek, 1947). Rogers (1952) discovered that 
ferrites have an extremely high Faraday’s effect, a feature fitting them 
for several applications in the microwave area.  When ferrites are 
employed in microwave devices, they are operated in the saturated 
state. The permeability of magnetically saturated ferrites in the 
microwave range varies with the saturation magnetization, the 
microwave frequency, and the outside magnetic field. Consequently, 
ferrites make tuneable photonic band gaps probable with functional 
fields. Kee et al. (2000) investigated theoretically the properties of 
two-dimensional ferrite photonic crystals for two independent 
polarizations. 
The propagation of magnetostatic waves in layered structures 
consisting of ferrite materials has attracted much The propagation of 
magnetostatic waves in layered structures consisting of ferrite 
materials has attracted much attention (Damon, 1961; Bongianni, 
1972; Bestler, 1959; Courtois, 1970; El-Khozondar, H. 2010), owing 
Magnetostatic Surface Waves Propagation at the interface 
 Journal of Al Azhar University-Gaza (Natural Sciences), 2013, 15           (61)  
to various applications of ferrite in the microwave devices and 
extremely important for designing integrated devices such as narrow-
frequency optical or microwave filters and high-speed switches 
(Vasseur, 1996; Al-Wahsh, 1999; Gulyaev, 2001). 
The type of surrounding materials affects the properties of the 
propagating modes. Therefore, it is interesting to see the combined 
effect of these materials on the surface wave propagation. In this 
paper, the transverse electric (TE) wave propagation in a structure 
consists of MTM film surrounded by a metal substrate and a 
magnetized ferrite cover bounded by air is investigated. 
The paper is organized as follows: In the following s ection, the 
theoretical steps are summarized. Next, the numerical results are 
discussed. Last section is dedicated for conclusion. 
2. Theoretical Analysis 
Figure 1 exhibits the configuration of the proposed waveguide 
which is an asymmetric planar waveguide. It consists of MTMs film 
having width w placed on a metal substrate and bounded from above 
by a Ferrite film with width s covered by air. 
 
      
Figure 1. Proposed asymmetric waveguide structure 
 
The slab has infinite extent in the z and y directions. We assumed 
lossless MTM film with both permittivity εM and permeability µM are 
function of the frequency (ω) as follows: 
2
2
1 pM
ω
ε
ω
= − ,                                                         (1) 
Zeyad I.Al-Sahhar, et al. 
(62)         Journal of Al Azhar University-Gaza (Natural Sciences), 2013, 15 
 
2
2 2
1M
r
Fω
µ
ω ω
= −
−
,                                                 (2) 
where ωr is the resonance frequency and ωp is the plasma frequency. 
The values of the parameters ωr, ωp and F are chosen to fit the 
experimental data (Shebly, 2001): ωp = 26.6π GHz, and F = 0.37. 
In the ferrite slab a static magnetic field is applied in the z 
direction, resulting in a uniform intensity H0 within the ferrite. The 
ferrite slab has positive permittivity εf and permeability µf defined as 
(Bestler, 1959; Courtois, 1970; El-Khozondar, H., 2010, Vasseur, 
1996). 
11 12
12 11
0
0 ,
0 0
f
z
i
i
µ µ
µ µ µ
µ
 
 = − 
  
                                   (3) 
where 11 2 21
M H
H
ω ω
µ
ω ω
= +
−
, 12 2 2
M
H
ωω
µ
ω ω
=
−
µz =1, ω is surface wave 
frequency, ωM =4πγM0  is the magnetic frequency, ωH = γH0 is the 
Larmor frequency, γ is the electromagnetic oscillation frequency, 
4πM0  is the ferrite saturation magnetization, and H0 is the applied 
magnetic field (Bespyatykh, 2001). 
In this study, we only considered transverse electric fields (TE). 
The TE fields are assumed to be E = (0, 0, Ez) e
jωt and H = (Hx, Hy, 0) 
ejωt. Waves are assumed to propagate in the y direction. The variation 
in the z-direction is assumed to be zero. Applying the defined fields 
into Maxwell's equations gives the following field equation, 
2 2
2
2 2
0,zi zi i zi
E E
q E
x y
∂ ∂
+ + =
∂ ∂
,                                   (4) 
where i indicates f for Ferrite layer, M for MTMs layer, and l for the 
linear cladding layer (air), 0lq k= , 0f f vq k ε µ= , 0M Mq k ε= , k0 = 
ω/c and 
2 2
11 12
11
v
µ µ
µ
µ
−
= , which is called the Voigt permeability. 
Solving above equations for each layer and implementing boundary 
conditions, we get the following dispersion equation, 
Magnetostatic Surface Waves Propagation at the interface 
 Journal of Al Azhar University-Gaza (Natural Sciences), 2013, 15           (63)  
( ) ( )
( )
2tanh
tanh 0,
fxk s
xf y v xl y xf xM v xM
xf y v xl y xf xM v xM
k k k k k k w k e
k k k k k k w k
ν ν
µ µ
µ µ
ν ν
µ µ
µ µ
−    + − − + +    
    
    
+ + + + =    
    
 (5) 
where 2 2 2y xi ik k q− = . The dispersion relation equation (5) relates the 
transverse wave numbers for each media. It is an implicit equation that 
gives the surface wave dispersion relation. When w→∞, equation (5) 
becomes, 
( )2
0.
fxk s
xf y v xl y xf v xM
xf y v xl y xf v xM
k k k k k k e
k k k k k k
ν ν
µ µ
µ µ
ν ν
µ µ
µ µ
−    + − − + +    
    
    
+ + + + =    
    
          (6) 
In the limit w = 0, the structure reduces to ferrite-metal structure and 
equation (5) reduces to the following form, 
( )coth 0.xf xf v xlk k s k k
ν
µ
µ
− + =                                          (7) 
Equation (5) to equation (7) can only be solved numerically. The 
solutions of these equations give the MSSW at the different limits. 
The limiting frequencies for MSSW free ferrite film are as follows 
(Damon, 1961): 
2
s H H Mω ω ω ω= +                                                         (8) 
/ 2fin H Mω ω ω= +                                                           (9) 
where ωs is the starting frequency and ωfin is the final frequency. 
3. Numerical Calculation 
In all the calculations, we chose ωH =1.76x10
7H0 rad/s, ωM 
=1.76x1870H0 rad/s, H0=367Oe, ε2=15, and ωp=83.56x10
9 rad/s. The 
value of frequency ω is chosen such that εM, µM, and µv are negative. 
Accordingly, ωs/ ωp =0.19 ,  ωfin/ ωp=0.27 and ωfinFM/ ωp=0.47. 
The normalized MSSW frequency with respect to plasma 
frequency (ω/ωp) is plotted as a function of propagation constant (ky) 
in Figure 2. The parameter values are: MTM-layer thickness 
w=79x10-4 cm and the ferrite thickness s varies as follows: s1=6.2 µm, 
s2=7.2 µm, s3=8.2x10
-4 cm and s4=9.2 µm. One can see that the curve 
Zeyad I.Al-Sahhar, et al. 
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is not distinct from the one for the free ferrite where MSSW appears at 
almost the same range of frequency interval, 0.1965 to 0.2333. 
However, there is not a second curve that exist for free ferrite film and 
would be symmetric to the existing curve regarding to the frequency 
axis. That is the MSSW is always directed oppositely to y-axis, i.e. 
this MSSW is unidirectional and backward.  
Figure 3 illustrates the dependence of the normalized MSSW 
frequency on ky at w=7.9 µm for the same values of s as mentioned 
above. We noticed that the MSSW can travel in both ± y-axis. 
However, the symmetry that exists in the free ferrite is lost. The 
forward MSSW takes place for the range of frequencies from 0.206 to 
0.219 and backward frequencies appears at the other frequencies. It is 
obvious from figure 2 and figure 3 that the behaviour of the MSSW 
varies with the changes of thicknesses of both the MTM layer and the 
ferrite layer.   
 
 
Figure 2. The normalized frequency versus ky at w=79 µm for different 
values of s.   
 
 
   s4 
   s3 
+  s2 
   s
Magnetostatic Surface Waves Propagation at the interface 
 Journal of Al Azhar University-Gaza (Natural Sciences), 2013, 15           (65)  
Similar calculation is done for s=7.9 µm and variant values of w as 
follows: w1=6.2 µm, w2=7.2 µm, w3=8.2 µm, and w4=9.2 µm. The 
normalized frequency versus ky is shown in figure 4. It is shown in the 
figure that both forward and backward MSSW exist and the symmetry 
regarding frequency axis is lost.  The forward MSSW occurs at 
frequencies rages from 0.206 to 0.219. Looking at figure 3 and figure 
4, we notice that the backward MSSW is clearly diverse for different 
values of s; on the other hand, the forward MSSW differs for different 
values of w. 
 
 
Figure 3. The normalized frequency versus ky at w=7.9 µm for 
different values of s.    
 
 
 
 
 
 
   s4 
   s3 
+  s2 
   s1 
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Figure 4. The normalized frequency versus ky at s=7.9 µm for different 
values of w.  
 
 
 
Figure 5. The normalized frequency versus ky at s=79 µm and different 
values of w.  
 
w4 
 
w
   w4 
   w3 
+  w2 
   w
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 Journal of Al Azhar University-Gaza (Natural Sciences), 2013, 15           (67)  
 
The MSSW normalized frequency dependence on propagation 
constant is plotted in figure 5 for constant value of s=79 µm and the 
values of w vary as follows: w1=6.2 µm, w2=7.2 µm, w3=8.2 µm, and 
w4=9.2 µm. It is observed from figure 5 that the MSSW changes as w 
changes and has different values for the forward modes. It is also 
found that the forward MSSW appears in two regions: in the first 
region the values of ω/ωp  change from 0.198 to 0.205 and in the 
second region the values of ω/ωp vary from 0.217 to 0.218.  
 
  
Figure 6. The normalized frequency as a function of propagation 
constant when w→∞ for different values of s. 
 
Figure 6 exhibits the relation between ω/ωp and ky when w→∞ for 
the following values of s: s1=6.2 µm, s2=7.2 µm, s3=8.2 µm, and 
s4=9.2 µm. It is shown that only backward MSSW appears at limited 
range of frequency ranges from 0.2294 to 0.2329. This behaviour 
   s4 
   s3 
+  s2 
   s1 
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agrees with the results we obtained in figure 2 and disagrees with the 
results in figure 3 where w decreases one order of magnitude. That is 
for small values of w both forward and backward MSSW exists while 
for large values of w only backward MSSW exists. 
The normalized magnetostatic waves frequency is plotted as a 
function of propagation constant at w = 0 for s varies as follows: 
s1=6.2 µm, s2=7.2 µm, s3=8.2 µm and s4=9.2 µm in figure 7. In this 
case only backward MSSW appears for the same range of frequencies 
from 0.1965 to 0.2333 as shown in figure 2 and changes as s varies. 
 
 
 
Figure 7. The normalized frequency as a function of propagation 
constant for w = 0 and different values of s. 
 
4. Conclusion  
The considered configuration is an asymmetric slab waveguide 
consisting of MTM film placed on a metal substrate and bounded 
from above by a Ferrite film covered by air. The dispersion relation 
for the magnetostatic waves are studied at three different values of the 
   s4 
   s3 
+  s2 
   s
Magnetostatic Surface Waves Propagation at the interface 
 Journal of Al Azhar University-Gaza (Natural Sciences), 2013, 15           (69)  
MTM layer thickness. The three resulted dispersion equations are 
numerically solved. Results are presented by plotting the MSSW 
frequency as a function of the propagation constant. Results 
demonstrate that MSSW frequency depends on both MTM layer and 
ferrite layer thicknesses. It is found that the range of the allowed 
frequency changes; moreover, the dependency of the positive and 
negative propagation constants change with changing MTMs 
thickness and /or the ferrite thickness. This result is promising in 
improving the waveguide performance. It also has industrial 
applications; i. e. isolators. 
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Magnetostatic surface waves propagation at the interface between ferrite and MTMs parallel plate waveguide structure

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Magnetostatic surface waves propagation at the interface between ferrite and MTMs parallel plate waveguide structure

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