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

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 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 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 b1 and b2 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:

(1)

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

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

(2)

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

(3)

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 ( propagation in y direction) versus frequency for a structure with p=12.7 mm, l=4.8mm and er=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 er. That is

In this periodic environment electric and magnetic fields are also periodic. So we have

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

Since we can conclude that

Writing this equation in matrix form we have:

(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:

Eliminating Hz from above equations and substituting equation (4) in it, gives the following equation:

(5)

In which A is a diagonal matrix with diagonal elements aii=-(2pi/p+b). To have a non-zero solution for Ex the determinant of above equation must be zero and therefore b could be calculated. Figure 4 shows the calculated b 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.

6. References

[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 on Microwave Theory and Techniques, Vol. 47, No. 11, Nov. 1999, pp. 2131-2138.

[3] Robert E. Collin, “Fundamental Foundation for Microwave Engineering”, 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 and Applications”, 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 Boundary Conditions 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 on Antenna 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 two dimensions”, Appl. Phys. Lett., vol. 61-64, no. 27, July 1992, pp. 495-497.