A Simple Method to Calculate the Propagation Constant
of a One-Dimensional Photonic Crystal
Nasrin Hojjat
Mahmoud Shahabadi
Assistant Professor
Assistant Professor
nhojjat@chamran.ut.ac.ir
Abstract
Periodic structures, some of which are known as
photonic band-gap (PBG) crystals, offer new dimensions
of freedom in controlling the behavior of electromagnetic
wave circuit and antenna[1]. PBG improves performance
of microstrip antenna and microwave circuit when etched
on high dielectric constant substrate material by
suppressing the suface waves[2]. In this paper we
calculate and compare the propagation constant of a onedirecsional Photonic Band-gap crystal by transmission
line and Fourier series expansion methods. In the first
method each unit cell of the structure will be modeled as
a transmission line and then applying the Floquet
theorem [3] the propagation constant of the structure will
be calculated. In the second method the propagation
constant of the same structure will be calculated by
Fourier series expansion of dielectric constant [4]. The
similarity between the two methods is very good.
1. Introduction
Electromagnetic Band-gap (EBG) structures are 3-D
periodic objects that prevent the propagation of the
electromagnetic waves in a specified band of frequency
for all angles and all polarizations states. However, in
practice, it is very hard to obtain such complete band-gap
structres and partial bang-gaps are achieved. Photonic
Band-gap crystals are one of the classes of EBG
structures which typically cover in-plane angles of arrival
and also sensitive to polarization states [5]. Various
numerical methods were used to study the wave
propagation in Photonic Band-gap crystals such as finite
difference in the frequency domain[6], finite difference in
the time domain[7], finite element method[8], plane wave
expansion method[9] and so on. In this paper we
represent two method for calculating the propagation
constant of a PBG structure. In the first method each unit
cell of the structure will be modeled as a transmission line
and then by the help of Floquet theorem the propagation
Shaghik AtaKaramians
Ms Student
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constant of the structure will be calculated. This method
which is called as transmission line method will be
introduced in section 2. In the second method the
propagation constant of the same structure will be
calculated by Fourier series expansion of dielectric
constant. This method is named Fourier Series Expansion
method and will be introduced in section 3.
2. Transmission line method
Assume a one dimensional periodic structure with
period ‘p’ as shown in Figure 1. The dielectric layer
width is ‘l’. The structure is infinite and homogenous in
the xz plane and we assume that electric field has no
component in the z direction. We could model the unit
cell of this periodic structure by two transmission line
cascaded together. The propagation constants of these
lines are  and  as shown in Figure 2. We assume that
propagation mode is TEM and we model the Electric and
Magnetic fields (Ex Hz) of the structure with voltage and
current of the transmission line equivalent circuit. From
the transmission line theory the voltage and current V3
and I3 in terms of V1 and I1 could be written as:
V3   A B  V1 
 I   C D  I  (1)
 1 
 3 
in which A, B, C and D are as follow:
x
r
0
r
0
y
l
p
p
Figure 1. One dimensional periodic structure
A  cos  0 ( p  l ) cos 1l 
In this periodic environment electric and magnetic
fields are also periodic. So we have
Z0
sin  0 ( p  l ) sin 1l
Z1
B   jZ1 sin 1l cos 0 ( p  l )  jZ0 cos 1l sin 0 ( p  l )
1
1
C   j sin  0 ( p  l ) cos 1l  j sin 1l cos  0 ( p  l )
Z0
Z1
Z1
D  cos  0 ( p  l ) cos 1l  sin  0 ( p  l ) sin 1l
Z0
According to Floquet theorem we could represent V 3
and I3 in the terms of V1 and I1 as :
V3  e y
I   
 3  0
0  V1 
 
e y   I1 
A  e

 C
B
De
y
(2)
 V1 
    0 (3)
  I1 
j
j 
1
D y e 

p0
dy
in which  is the propagation constant in the y direction.
Since D   0 r E we can conclude that
 2m

   y
p

j 
1
Dm    0 r Ee 
p0
Dm   0
0
p
M
lim
M 
M
E
m   M

m   M
p
dy
Em   r  y e

m m  m
  

D 

0
 1 

 D0    0   1



2
 D1 

  


m  m 
y
j  2
p 

dy
0
  0 m  E m

 1
0
1
D  0N 2E
In this method we use Fourier series expansion method
to calculate the propagation constant of the periodic
structure represented in Figure 1. Since structure is
periodic in the y direction we can use Fourier series
expansion to expand r. That is
 me
 2m

   y
p

p
Dm 

 2
 1
0
  
E 
  1 
  E0 


  E1 
   
Writing this equation in matrix form we have:
3. Fourier series expansion method
M
 2m

 j 
   y
 p

M  m   M

To have a non zero solution the determinant of the
above matrix should be zero which gives the values for
 . Figure 3 shows propagation and attenuation constants
(     j ,   0 propagation in y direction) versus
frequency for a structure with p=12.7 mm, l=4.8mm and
r=9.
 r  y   lim
 Dm e
p
from equations (1) and (2) we can conclude:
y
M
D y   lim
(4)
Since the structure is infinite and homogenous in the
xz plane and we assume that electric field has no
component in the z direction, Maxwells equations are
simplified as follow:
2m
y
p
M  m   M
m 
r
p
j
1
 r  y e

p0
V1
I1
2m
y
p
dy
l
p-l
1
0
V3
I3
p
l
Figure 2. Transmission line model for one period
Attenuation and propagation constant
of the periodic structure in y direction
Figure 3.
M=3
TLM
Figure4.
Attenuation and propagation constant of the periodic structure
E X
 jH Z
y
H Z
 jDX  j 0 N 2 E X
y
Eliminating Hz from above equations and substituting
equation (4) in it, gives the following equation:
A    
2
0 0

N 2 EX  0
(5)
In which A is a diagonal matrix with diagonal elements
ii=-(2ip+. To have a non-zero solution for Ex the
determinant of above equation must be zero and therefore
 could be calculated. Figure 4 shows the calculated  for
the same structure compared with transmission line
method. The close agreement between the results is
observed even for small numbers of Fourier harmonics
(M=3). It is anticipated that by increasing M, the results
become more similar.
4. Conclusion
In this paper two different methods for calculating the
propagation constant of a periodic structure are compared.
For a 2-D structure the same two methods could be
applied to calculate the propagation constant. This
application will be considered in a separate paper. For a
one-dimensional structure the transmission line method is
apparantly the most efficient method both in terms of the
accuracy and computation time, while for a 2-D structure
the Fourier expansion method is the method which could
find the propagation constant in all directions.
5. Acknowledgment
The authors would like to thank the Center of
Excellence of Applied Electromagnetics and the Electrical
and Computer Engineering Department of Tehran
University.
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Nov.
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