A Simple Method to Calculate the Propagation Constant

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A Simple Method to Calculate the Propagation Constant

of a One-Dimensional Photonic Crystal

Nasrin Hojjat Mahmoud Shahabadi Shaghik AtaKaramians

Assistant Professor Assistant Professor Ms Student

[email protected] [email protected]

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 one-

direcsional 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 theelectromagnetic 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. PhotonicBand-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]. Variousnumerical methods were used to study the wave

propagation in Photonic Band-gap crystals such as finite

difference in the frequency domain[6], finite difference inthe time domain[7], finite element method[8], plane wave

expansion method[9] and so on. In this paper werepresent two method for calculating the propagation

constant of a PBG structure. In the first method each unitcell of the structure will be modeled as a transmission line

and then by the help of Floquet theorem the propagation

constant of the structure will be calculated. This method

which is called as transmission line method will beintroduced 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 Expansionmethod 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 inthe 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 linecascaded together. The propagation constants of these

lines areβ1 and

β2 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. Fromthe transmission line theory the voltage and current V 3

and I3 in terms of V1 and I1 could be written as:

=

1

1

3

3

I

V

DC

B A

I

V (1)

in which A, B, C and D are as follow:

x

y

l

p

p

0ε

r ε

0ε

r ε

Figure1. One dimensional periodic structure

mailto:[email protected]






l l p Z

Z l l p A 10

1

010 sin)(sincos)(cos β β β β −−−=

)(sincos)(cossin 010011 l pl jZ l pl jZ B −−−−= β β β β

)(cossin1

cos)(sin1

01

1

10

0

l pl Z

jl l p Z

jC −−−−= β β β β

l l p Z

Z l l p D 10

0

110 sin)(sincos)(cos β β β β −−−=

According to Floquet theorem we could represent V3 and I3 in the terms of V1 and I1 as :

=

−

−

1

1

3

3

0

0

I

V

e

e

I

V y

y

γ

γ

(2)

from equations (1) and (2) we can conclude:

−−

−

=

−

−0

1

1

I

V

e DC

Be A y

y

γ

γ

(3)

To have a non zero solution the determinant of the above matrix should be zero which gives the values for γ . Figure

3 shows propagation and attenuation constants ( 0, >+= β β α γ j propagation in y direction) versus frequency for a

structure with p=12.7 mm, l=4.8mm and εr=9.

3. Fourier series expansion method

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 εr .That is

In this periodic environment electric and magnetic fields are

also periodic. So we haveFigure2. Transmission line model for one period

( )

( ) dye y p

1

e y

p

0

y p

m2 j

r m

M

M m

y p

m2 j

m M

r lim

∫

∑

=

=−=

−

∞→

π

π

ε ε

ε ε

( )∑ ∫

∫

−=′

′−

′∞→

+

=

=

M

M m

p y p

mm j

r m M

y p

m j p

r m

dye y E p

dy Ee p

D

0

2

0

2

0

0

lim

1

π

β π

ε ε

ε ε

l

l

p

1

1

I

V

1β 0

β r ε

3

3

I

V

p-l



( )

( ) dye y D p

D

e D y D

y p

m j p

m

M

M m

y p

m j

m M

+

−=

+−

∞→

∫

∑

=

=

β π

β π

2

0

2

1

lim

in which β is the propagation constant in the y direction.

Since E D r ε ε 0= we can conclude that

∑−=′′−′

∗== M

M mmmmmmm E E D ε ε ε ε 00

=

−

−

−−−

1

0

1

012

101

210

0

1

0

1

E

E

E

D

D

D

ε ε ε

ε ε ε

ε ε ε

ε

Writing this equation in matrix form we have:

E N D

2

0ε = (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:

(9-a)

Figure3. Attenuation and propagation constant of the

periodic structure in y direction



Z X H j y

E ωµ =

∂∂

X X Z E N j D j y

H 2

0ωε ω ==∂∂

Eliminating Hz from above equations and substituting

equation (4) in it, gives the following equation:

( ) 02

00

2 =+ X E N A ε µ ω (5)

In which A is a diagonal matrix with diagonal elements

αii=-(2πi/ p+β). To have a non-zero solution for E x the

determinant of above equation must be zero and therefore

β could be calculated. Figure 4 shows the calculated β for

the same structure compared with transmission line

method. The close agreement between the results isobserved 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 beapplied to calculate the propagation constant. This

application will be considered in a separate paper. For aone-dimensional structure the transmission line method is

apparantly the most efficient method both in terms of theaccuracy and computation time, while for a 2-D structure

the Fourier expansion method is the method which couldfind 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 TehranUniversity.

6. References

Figure4.Attenuation and propagation constant of the periodic structure

M=3TLM



[1] y. Qian and T. Itoh, “Planar Periodic Structures for Microwve and Millimeter Wave Circuit Applications”, IEEE,

MTT-S Digest 1999, pp. 1533-1536.

[2] R. Gonzalo, P. Magget and M. Sorolla, “Enhanced Patch-Antenna Performance by Suppressing Suface Wave using

Photonic Bandgap Substrates”, IEEE Transactions onMicrowave Theory and Techniques, Vol. 47, No. 11, Nov.

1999, pp. 2131-2138.

[3] Robert E. Collin, “Fundamental Foundation for MicrowaveEngineering”, Mc GrawHill 1966.

[4] M. Shahabadi, "Anwendung der Holographie auf Leistungsadition bei Millimeterwellen", Ph.D. dissertation,

Reihe 10 Informatik Kommunikations-technik, Hamburg, 1998.

[5] Y. Rahmat-Samii and H. Mosallaei, “Electromagnetic Band-

gap Structures: Classifications, Charactrization andApplications”, IEE, 11th International Conference on Antennas

and Propagation, April 2001, pp. 560-564.

[6] R.V.D. Espirito Santo, C.L.D.S. Souza Sobrinho and A.J.Giarola, “Analysis of 2-D Periodic Structures using the FD-FD

method”,IEEE 1998, pp. 402-405.

[7] A. Mekis, S. Fan, J.D.Joannopoulos, “Absorbing BoundaryConditions for FDTD Simulation of Photonic Crystal

Waveguides”, IEEE, Microwave and Guided Wave Letters, Vol.9, NO 12, December 1999, pp. 502-504.

[8] G. Pelosi, A. Cocchi and A. Monorchio, “A Hibrid FEM-

Based Procedure for the Scattering form Photonic Crystals

Illuminated by a Gaussian Beam”, IEEE Transactions onAntenna and Propagation, Vol. 48, NO 6, June 2000, pp. 973-980.

[9] R.D. Meade, K.D. Brommer, A.M. Rappe and J.D.

Joannopoulos, “ Existence of a Photonic Bandgap in twodimensions”, Appl. Phys. Lett., vol. 61-64, no. 27, July 1992,

pp. 495-497.

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A Simple Method to Calculate the Propagation Constant