The Islamic University Journal (Series of Natural Studies and Engineering)
Vol.15, No. 1, pp 147 -155 , 2007, ISSN 1726-6807, http// www.iugaza.edu.ps/ara/research
Nonlinear Electromagnetic TM Surface Waves in Magnetic
Superlattices(LANS)Film
Dr.H.M.Mousa
Physics Department, Al.Azhar University, Gaza,
Gaza Strip, Palestinian Authority,
E.mail: h.mousa@alazhar-gaza.edu
Abstract The propagation characteristics of nonlinear TM surface waves at a lateral
antiferromagnetic/nonmagnetic superlattices (LANS) film and a nonlinear dielectric cover
have been investigated. LANS are linear frequency- dependent gyromagnetic media. They
are described with an effective- medium theory. It is found that the frequency – wave index
variation increases with the magnetic fraction f1 . We also calculate and illustrate the
variation of the wave index with the power flow for various values of
f1 . We found that
f1 increases the power of the nonlinear TM surface waves.
Keywords: Nonlinear Waves, Wave-guides, Dispersion relation, Magnetic Superlattices.
1. Introduction
During the last few years, nonlinear behavior of electromagnetic waves in
antiferromagnetic films have attracted a significant degree of attention1. At
the present time little seems to be known about solutions of Maxwell’s
equations that describe the propagation of surface or guided waves in
nonlinear structures that involve linear gyromagnetic media. In addition
almost all of the exact studies of TM nonlinear surface waves have been
based on frequency – independent dielectric constants and attention has
focused upon the infrared region of the spectrum.
Boardman and Shabat et al2 had studied nonlinear TM surface waves along
a single interface of a linear ferrite substrate and nonlinear magnetic
cladding. They found that TM waves can propagate even if such
propagation is in the linear, low-power medium. Wang et al3 also studied
nonlinear TM surface waves on a structure of antiferromagnetic and linear
ferromagnetic media. They found that the interface could supports the
nonlinear TM surface waves. Hamada et. al4 studied nonlinear TM surface
waves on a structure of nonlinear antiferromagnetic medium and linear
superconductor substrate. They found that the variation of the frequency and
power flow of TM waves with wave index is temperature dependent.
In this paper, we investigate the propagation characteristics of nonlinear
surface waves at LANS superlattice film which are described with an
Nonlinear Electromagnetic TM Surface Waves in Magnetic
effective medium theory. Such description is valid when the wave length of
the excitations are much longer than the superlattice period where kL  1,
where k is the magnitude of the wave vector and L=L1+L2, is the period of
the superlattice, L1 and L2 are the thickness of the antiferromagnetic layers
and non-magnetic layers, respectively 5 .
The magnetic fractions of the LANS superlattices are introduced as:
L
L
f1  1 and f2= 2 ,
L
L
and are called the magnetic and non-magnetic fractions respectively, where
f1  f 2  1 .
y
Nonlinear cover  nl  3
t
0
Superlattices  e  e
x
Linear dielectric  4
Fig..(1 ) TM surface waves waveguide composed of ( LANS)
layered structure .
2. Basic Equations
The guiding structure considered here is shown in Fig. (1). In this structure
a superlattice film (LANS) of finite thickness (t) is sandwiched by a semi
infinite nonlinear cladding y > t and a semi infinite linear dielectric substrate
in the region y < 0. The effective dielectric tensor of (LANS) is described
as6 :
148
Dr. H. M. Mousa
 

 e  0
0

0
 ||
0
0

0
  
Where
    1 f1   2 f 2
(1a)
1  2
 || 
(1b)
 1 f 2   2 f1
The electric and magnetic field vectors for TM waves propagating along xaxis in the xy- plane with an angular frequency  and a wave vector k x
(2)
will take the form:

H  0 , 0, H z ( , z )exp i k x x   t 

(3)
E  E x ( , y ) , E y ( , y ), 0exp i k x x   t 
In the presence of the infra-red field associated with a TM wave propagating
along the interface , the non-linear permeability of an isotropic magnetic
cladding is given by
(4)
 N L   L   H z2
This expression arises from an expansion of the permeability about the
applied static field H o . Hence H z is the ac magnetic field carried by the
TM wave,  L is the linear part of the permeability and  is the non-linear
coefficient.. H z is also real because only stationary, non-radiating waves
will be considered7,8 .
The wave equation in each layer is obtained from Maxwell’s equations:


  E  i   0  NL H
(5)


(6)
 H   i  0  3 E
where  3 is the relative permitivity. Substituting equations (2),(3) into (5)
and equation (6) yields the following three differential equations in the three
layers:
2 H z
 k 32  k 02  3  H z2  H z  0
,
y>t ,
2
(7)
y

 2 H z    2
2


k

k

0
 H z  0 ,
 x
 y2
 ||

2
 Hz
2
 k x  k 02  4 H z  0 ,
2
y


149
0 y t,
y< 0 ,
(8)
(9)
Nonlinear Electromagnetic TM Surface Waves in Magnetic
kx 2  2
, k 0  2   o  o  2 ,  o and  o are the dielectric permitivity
k0
c
and magnetic permeability of free space respectively .
An appropriate solution of equations has the form2:
1- In nonlinear cover:
1
2
(10)
Hz 
k 3 sec h k 3  y  y 0  ,
k0   3
where n x 


where k3  k 0 nx2  k 02 3 l ,
(11)
and y0 is a constant of integration that defines the position of a self focused
peak in H z .
2. In superlattice (LANS)layer
(12)
H z  A1 sinh(  y )  A2 cosh(  y )
where   k 0
(13)
 2
nx   
 ||
3.In linear dielectric substrate
H z  Ce k1 y
(14)
where k1  k 0 n x2   4
(15)
.
Here, A1 , A2 are the field amplitudes in (LANS) and C is the field
amplitude in the dielectric which can be determined by the boundary
conditions.
By requiring the tangential components of E x and H z at the boundary y =
t as:
150
Dr. H. M. Mousa
k 32
 3k0
,
k3
k0
2
3
sec h k 3 (t  y0 )  tanh k 3 (t  y0 )  
2
3

A1 cosh(  t )  A2 sinh(  t )

sec h k3 (t  y0 )   A1 sinh(  t )  A2 cosh(  t ),
Continuity of H z
(16)
(17)
and E x at y = 0, yields the following equations:
(18)
 A1 k1C

,

4
A2  C ,
Equations (18) and (19) can then yield
k
A1  1  A2
(19)
(20)
4
By dividing Eq.(16) over Eq.(17) and using Eq.(20), the dispersion equation
is then obtained as:
 (k 3 4   v  k1 3  )
tanh(  t ) 
 2 4 3  k1 k 3 2 v
(21)
where v  tanh k 3 ( y 0  t ) ,
is called the magnetic nonlinearity
TM waves power flux
The total power flux(p) of the waves propagating along the x direction is:

p
1
E z H y dy
2 
(22)
 p NL  psup  p die ,
where p NL , psup and pco are respectively the power fluxes in the nonlinear
cover, superlattice and dielectric media and given by:
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Nonlinear Electromagnetic TM Surface Waves in Magnetic
p NL 
psup
2k x k 3
 0 32 k 02
1  tanh( k 3 (t  y0 )) ,
(23a)
 k   2  cosh(  t ) sinh(  t )   t 
k1 
1 






2




2
 4
k A2   4   

 x 2 
 0  II  cosh(  t ) sinh(  t )   t 



2


p die 
 sinh 2 (  t  

  
 2   ,
 (23b)



k x A22
(23c)
2 0  4 k1
From
Eq.(17)
obtains A  2
1  v  k1
k 0  3
 4
2
2
k 32
2
one
2
(23d)


 sinh(  t )  cosh(  t ) ,


3. Numerical results and Discussion
To compute the dispersion curves directly, we first solve the dispersion
equations numerically, this is done by fixing the parameter y 0 which is the
location of the maximum in the non-linear function, given by equation (10)
then roots of equation (21) are found by varying n x which is chosed
according to the following conditions:
and n x   L  3 n x   3  ||
Numerical calculations for dispersion curves are found, examples of the
dispersion curves are computed for a lateral FeF2 / ZnF2 super lattice and
non-linear material consists of a suspension of short graphite fibers in
heptane and oil. We take the parameters as follows5:
m0  0.56 k G, H a  200 k G, H e  540 k G ,   1.97 10 7 rad / sec .G and  1  5.5 for
antiferromagnetic
layers,
2  8
for
the
nonmagnetic
layers,  3  2.25 ,  L 1.29 for the non-linear medium and  4  3 for the
substrate4.
The propagation of the TM surface waves is reciprocal where
 (k x )   ( k x ) . The frequency – wave index variation for different values
of the magnetic fraction f1 is demonstrated in Fig.(2). It shows optical
instability behavior9 i.e. for curve of label (1) where f1 = 0.4, it shows that
for the wave angular frequency  =1*1016 rad /sec there are two values of
152
Dr. H. M. Mousa
n x (2.8, 2.9). The optical instability is affected by the magnetic fraction f1
where the wave velocity increases by increasing f1 .
The frequency- wave index variation for different values of the magnetic
nonlinearity is shown in Fig.(3). It displays that the wave frequency is
affected by the magnetic nonlinearity . The wave frequency increases by
decreasing the magnetic nonlinearity
Once the propagation characteristics are determined from the dispersion
equation (21), the obtained values of the refractive index can be fed to the
power expression mentioned in Eq. (22).As illustrated in Fig.(4) the
normalized P P0 has been plotted against n x for different values of f1 . It
illustrates the dependence of the normalized power on the magnetic fraction
where the increasing of the magnetic fraction causes increasing of the
power. Fig (5) shows a typical field distribution H Z as a function of the
distance from the interface . If the wave index is increased, however, the
maximum of the field is established at a smaller distance from the interface
y = 0.
Fig..2.
Dispersion curves of TM surface waves for
H0=.2kG, and v= 0.16, t =0.44  10 7 m (1) f1 = 0.4, (2) f1 =
0.6, and (3) f1 = 0.8. The curves are labelled with values of
α = 1.55  1010 m2 V-2 ,  l =1.29 ε3 = 2.25, ε1 =5.5, ε2 = 8 and
ε4= 3.
153
Nonlinear Electromagnetic TM Surface Waves in Magnetic
Fig.3 Dispersion curves of TM surface waves for
H0=.2kG, and f1= 0.8, t =0.44  10 7 m (1) v = 0.25,
(2) v = 0.2, and (3) v = 0.16. The curves are labelled
with values of α = 1.55  1010 m2 V-2 ,  l =1.29 ε3 =
2.25, ε1 =5.5, ε2 = 8 and ε4= 3.
Fig. 4. Normalized TM waves power flow along the
x-direction as a function of wave index ( n x ) for t
= 0.44  10 7 m, v = 0.16 (1) f1 = 0.5, (2) f1 =0.6, (3)
f1 = 0.7.
154
Dr. H. M. Mousa
Fig. 5. Variation of TM field component across LANS
film for , f1 = 0.8, t = 0.44  10 7 m, v = 0.16 (1 )nx =2.482
 0.914 x 1016 rad/s (2) nx =2.51  =0.1084 x 1017 rad/s
and (3) nx =2.55,  0.245 x 1017 rad/s, the other data
as in Fig .2.
4. Conclusions
The TM power flow is dependent on the magnetic fraction. Magnetic
fraction increases the power levels needed to observe strong nonlinear
waves. By increasing the wave index, the magnetic field distribution
concentrates near the interface in the nonlinear medium. We believe that the
carried work will lead to future promising application in microwaveinfrared technology.
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