The nonlinear characteristics of magnetostatic surface waves at microwave frequencies in a layered structure of left-handed material film and a semi-infinite linear ferrite substrate have been investigated. The general dispersion relation is derived and analyzed numerically. It is found that it has two solutions for ω (k), one represents a physical solution and other unacceptable. The effects of the applied external magnetic field around the proposed region have also been examined

Hamada, Majde S.

Mansour, Shawki A

Shabat, Mohammed M.

English

Institutional Repository of the Islamic University of Gaza

 
 
 
Sh. Mansour et al., J. Al-Aqsa Unv., 10 (S.E) 2006 
 
 263
Magnetostatic surface waves on left-handed materials (LHM) 
Dr. Shawki A. Mansour  *  
Prof. Mohammed M. Shabat * 
Dr. Majdi S. Hamada ∗∗  
  
 
ABSTRACT 
 
The nonlinear characteristics of magnetostatic surface waves at 
microwave frequencies in a layered structure of left-handed material film 
and a semi-infinite linear ferrite substrate have been investigated. The 
general dispersion relation is derived and analyzed numerically. It is found 
that it has two solutions for ω (k),one represents a physical solution and 
other unacceptable. The effects of the applied external magnetic field 
around the proposed region have also been examined. 
 
Keywords: Left-handed materials; Dispersion relation; Magnetostatic 
surface waves; wave-guides; Boundary method. 
 
 
 
 
 
 
 
 
 
 
 
                                                           
* Physics Department, Faculty of Applied Sciences, Islamic University of Gaza, P. 
O. Box 108, Gaza, Gaza Strip, Palestine, E-mail: shmansour@mail.iugaza.edu  
∗∗ Physics Department, Faculty of Applied Sciences, Al-Aqsa University, P. O. 
Box 4051, Gaza, Gaza Strip, Palestine. 
 
 
 
 
Magnetostatic surface waves on … 
  
 264
INTRODUCTION: 
Left-handed material (LHM)1-7 have been receiving an increasing interest 
because of their proposed applications in future microwave engineering 
technology. Recently, Shelby et al1 have demonstrated a two dimensionally 
isotropic left-handed material which consist of a two dimensionally periodic 
array of copper split ring resonators and wires (SSR). The LHM are at 
certain band of frequencies, behaves with negative ε(ω) and µ(ω), and also 
have imaginary part, thereby the refractive index of such metamaterial 
exhibits a negative value in this frequency as produced by Veslago2. A 
LHM also verify some of the explicit prediction such as reversed refraction, 
backward, Cherenkove radiation and reversed Doppler effect3.  The study of 
nonlinear optical effects in the various waveguides structures containing a 
gyromagnetic media such as a ferrite8 is also considered a key problem of 
the simulation of a number of opto-microwave electronic devices. 
 
 
 
 
 
 
 
Fig. 1. Coordinate system for the single interface between LHM and a 
linear ferrite  cladding, the applied magnetic field is in the Y-direction. 
 
In this paper, we investigate the propagation characteristics of nonlinear 
magnetostatic surface waves at microwave frequencies in a layered structure 
of left-handed metamaterial cover and a linear ferrite substrate as shown in 
fig. 1. 
Basic Equations: 
The guiding structure to be considered consist of a LHM film which 
characterized by6: 
( )
2
2 2
0
1eff
Fωµ ω
ω ω
= −
−
    ,       ( ) 2
2
1
ω
ω
ωε epeff −=           (1) 
Where F = 0.56 , ω0 / 2π = 4 GHz , and ωp / 2π = 10 GHz 
And, a gyromagnetic ferrite (YIG) substrate is described by a magnetic 
permeability tensor as: 
Linear Ferrite cladding 
(semi-infinite) 
L.H.M. cover 
(µ,ε) < 0 
X 
Z 
(2) 
(1) 
 
 
 
Sh. Mansour et al., J. Al-Aqsa Unv., 10 (S.E) 2006 
  
 265
( )
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
=
xxxz
B
xzxx
µµ
µ
µµ
ωµ
0
00
0
      (2) 
Where, 
( )
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−+
= 22
0
2
00
ωω
ωωωωµµ mBxx ,    22
0 ωω
ωωµµ
−
= mBxz i     (3) 
and µB is the usual Polder tensor elements, ω is the angular frequency of 
the supported wave, ω0 = γµ0H0 , ωm = γµ0M0 , H0 is the applied magnetic 
field, γ = 1.76×1011 S-1T-1 is the gyromagnetic ratio, M0 is the dc saturation 
magnetization of the magnetic insulator and µB has been introduced as the 
background, optical magnon permeability. 
The electric and the magnetic field of TE wave propagating in the x-
direction can be written as: 
( ) ( )[ ]ctzikEE y −= β0exp0,,0        (4) 
( ) ( )[ ]ctzikHHH zx −= β0exp,0,        (5) 
Where 
0k
k
=β  is the complex effective wave index constant, k0 is the wave 
number of the free space, and c is the velocity of light in free space. 
In the ferrite substrate: 
The magnetostatic potential Ψ of the magnetostatic surface waves in the 
YIG film is written 8 as: 
( ) ( ) ( )tkxiekxA ω−=Ψ exp1        (6) 
The relevant component of the magnetic fields for the TE magnetostatic 
waves in the YIG can be written after considering the phase difference as: 
( ) ( ) ( )tkxix ekzikAh ω−= exp1        (7) 
( ) ( ) ( )tkxiz ekzikAh ω−−= exp1          (8) 
( ) ( ) ( )( )1101 zxxxxzy hhSke µµ
ωµ
+−=
       (9) 
Where, 1±=S , S = 1 stands for the propagation of the waves in forward 
direction, and S = -1 for the backward direction. 
In LHM-cover: 
Maxwell's Equations are: 
( )HiE eff ωµωµ0=×∇       (10) 
( )EiH eff ωεωε 0=×∇        (11) 
 
 
 
Magnetostatic surface waves on … 
  
 266
Where the effective permeability and the effective permittivity both are 
less than zero. 
From Eq. (10) we get: 
( )
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
∂
∂
z
x
eff
y
H
H
i
E
z
ik
kji
0
00
0
ˆˆˆ
0 ωµωµ       (12) 
The components of the electric field and magnetic field are: 
xeff
y Hi
z
E
µωµ0=∂
∂−
        (13) 
Then we get: 
z
EiH y
eff
x ∂
∂
=
µωµ0
                 (13a) 
Similarly, 
zeffy HiikE µωµ0=         (14) 
Then we get: 
y
eff
z E
kH
µωµ0
=                (14a) 
Applying Eq. (11) the components of magnetic field is: 
( )0
ˆˆ ˆ
0
0
00
eff y
x z
i j k
ik i E
z
H H
ωε ε ω
⎛ ⎞
⎛ ⎞⎜ ⎟
∂ ⎜ ⎟⎜ ⎟ = − ⎜ ⎟⎜ ⎟∂ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟
⎝ ⎠
      (15) 
Then we get: 
0
z
z eff y
HikH i E
z
ωε ε∂− + = −
∂
       (16) 
Substitute both Eq's. (13a) and (14a) in Eq. (16) respectively, we obtain: 
yeff
y
eff
y
eff
Ei
z
Ei
z
Ekik εωε
µωµµωµ 000
−=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−    (17) 
Multiplying Eq. (17) by ωµ0µeff, we get: 
( ) 000222
2
=−−
∂
∂
yeffeff
y Ek
z
E
εµεµω       (18) 
 
 
 
Sh. Mansour et al., J. Al-Aqsa Unv., 10 (S.E) 2006 
  
 267
But 2
2
2
0 c
k ω=   where, 002
1 µε=
c
                (18a) 
And k = k0β                  (18b) 
Substitute both Eq's. (18a) and (18b) in Eq. (18), we obtain: 
( ) 0202202
2
=−−
∂
∂
yeffeff
y Ekk
z
E
εµβ        (19) 
Let ( )2 2 21 0 eff effk k β µ ε= −                  (19a) 
Finally, we got on a 2nd diff. eq. on the form:  
0212
2
=−
∂
∂
y
y Ek
z
E
         (20) 
And its dolution as: 
zk
y AeE 1=        and    effeffkk εµβ −=
2
01               (20a) 
Where, A is a constant  
The relevant components of magnetic fields and the electric field in LHM 
have the form: 
( ) ( )txkik
eff
x ee
iBkH ω
µωµ
−= 11
0
12         (21) 
( ) ( )txkizk
eff
z ee
AkH ω
µωµ
−= 11
0
12         (22) 
( )txkizk
y eBeE
ω−= 112          (23) 
With 21 0 eff effk k β µ ε= −                  (23a) 
But for TE-waves it can be shown that8 there is a 
2
π  phase difference 
between Hx and Hz. It is converted to redefine the field components as: 
hx = hx , hz = ihz and ey = iey , so the field components can be written in the 
left handed material cover as: 
( ) ( )txkizk
eff
x ee
iBkH ω
µωµ
−= 11
0
12         (24) 
( ) ( )txkizk
eff
z ee
iBkH ω
µωµ
−= 11
0
12         (25) 
( )txkizk
y eiBeE
ω−= 11          (26) 
Applying the boundary conditions for the continuity of tangential H at z = 0 
we get:  
 
 
 
Magnetostatic surface waves on … 
  
 268
( ) ( )21
xx Hh = , then we have:  
( ) ( )txki
eff
tkxi eBkkAe ωω
µωµ
−− = 1
0
1         (27) 
And,  ( ) ( )21 yy Ee = , then we have: 
( ) ( ) ( )txkitkxixxxz BeeSA ωωµµωµ −− =+− 10        (28) 
Dividing Eq. (27) by Eq. (28) we obtain: 
( )xxxz
eff
S
k
k µµ
µ
+
−
=
1
1
         (29) 
With,     effeffkk µεβ −=
2
1  and k = k0 β              (29a) 
Substitute Eq. (29a) in Eq. (29) and simplifying to get on the general 
dispersion equation. 
( ) ( )
( ) ( ) 2
2
effxxxzxxxz
xxxzxxxzeffeff
SS
SS
µµµµµ
µµµµµε
β
−++
++
=        (30) 
Numerical Results and discussions: 
Numerical computations is performed in order to calculate the propagation 
characteristics of the nonlinear dispersion equation. The numerical 
calculations were carried out with the same parameters for the ferrite (YIG) 
substrate as8. the data parameters for the LHM as5: F = 0.56 , ω0 / 2π = 4 
GHz , and ωp / 2π = 10 GHz. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Sh. Mansour et al., J. Al-Aqsa Unv., 10 (S.E) 2006 
  
 269
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 2. Dispersion curves in both forward, backward direction,  µ0H0 = 
0.5 T, µB = 1.25,  µ0M0 = 0.1750 T, γ = 1.76 ×1011 rad s-1 T-1, GHzp 10
2
=
π
ω , 
GHz4
2
0 =
π
ω , F  = 0.56. 
 
In this case, the frequency range in which both ε(ω) and µ(ω) are 
negative extends from ωo up to 6 (GHz), where the refractive index is 
expected to take a negative value. Moreover, Fig.2 shows the linear 
dispersion curves which is the variation of the frequency with the wave 
index are display the (i) expected reciprocal behavior, which is important in 
microwave signal processing technology, (ii) since the propagation 
characteristics both forwards to the right and symmetrical ( k>0), (iii) the 
frequency is decreasing with increasing the wave number k and this happen 
only in the propagation of TE-magnetostatic surface waves. Its also notice 
that, by increasing the applied external magnetic field, µ0H0 by 0.52T, 0.55T 
and 0.6T respectively, the TE surface waves disappeared in the proposed 
region, 4 10f≤ ≤  (GHz). For f < 4(GHz), where µv > 0 and ε < 0, the 
medium is transparentmedium and the guiding structure becomes a metallic 
as in Fig. 3, and for 6 < f < 10 (GHz) there is no physical solution while 
for 10f ≥ (GHz), where (µv > 0, ε > 0), the physical solutions are starting to 
appear and the media becomes a dielectric. 
   
 
 
 
 
 f G ) 
β
 ( Hz 
 
 
 
Magnetostatic surface waves on … 
  
 270
 
 
 
 
 
 
 
 
 
 
 
Fig. 3. Shows the Dispersion characteristics considered as a metal in 
case ( f < 4 GHz ), µ0H0 = 0.5 T, µB = 1.25, µ0M0 = 0.1750 T, γ =1.76 ×
1011rad s-1T-1, 10
2
p GHz
ω
π
= , 0 4
2
GHzω
π
= , F= 0.56 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 4. Shows the Dispersion characteristics considered as a dielectric in 
case    ( f > 6 GHz ), µ0H0 = 0.5 T, µB = 1.25, µ0M0 =0.1750 T, γ  =1.76 ×
1011 rad s-1T-1, 10
2
p GHz
ω
π
= , 0 4
2
GHzω
π
= , F= 0.56 
On the other hand, we increased the applied external magnetic field µ0H0 
by 0.2T and 0.3T for 4f ≤ (GHz), fig. 5 and for 10f ≥ (GHz), fig. 6, it is 
seen that the propagation in the forward direction began to decrease 
respectively.    
β
f (GHz) 
β
f (GHz) 
 
 
 
Sh. Mansour et al., J. Al-Aqsa Unv., 10 (S.E) 2006 
  
 271
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig.(5a-b). Shows the Dispersion characteristics in case ( f < 4 GHz ), 
µ0H0 = 0.2T, 0.3T respectively, µB = 1.25, µ0M0 = 0.1750 T, γ =1.76 ×1011 
rad s-1T-1, 10
2
p GHz
ω
π
= , 0 4
2
GHzω
π
= , F= 0.56 
 
 
 
 
 
 
 
 
 
 
 
Fig.(6a-b). Shows the Dispersion characteristics in case ( f ≥10 GHz ), 
µ0H0=0.2T, 0.3T respectively, µB = 1.25, µ0M0 = 0.1750 T, γ =1.76 ×1011 
rad s-1T-1, 10
2
p GHz
ω
π
= , 0 4
2
GHzω
π
= , F= 0.56 
CONCLUSION: 
A general dispersion equation that describes the propagation of 
magnetostatic surface waves in a YIG film, bounded by a left handed cover, 
has been derived theoretically. We proposed here an approach describing a 
new type of reciprocal behavior in a two-layer structure, which is very 
promising for opto-microwave electronic devices. 
 
 
β
f 
(GHz)
(a
β 
f 
(GHz)
(b
β
 
f 
(GH
β 
(b
f 
(GHz)
 
 
 
Magnetostatic surface waves on … 
  
 272
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Magnetostatic surface waves on left-handed materials (LHM)

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Magnetostatic surface waves on left-handed materials (LHM)

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