**5. References**


**Part 4** 

**Design and Modeling** 


**Part 4** 

**Design and Modeling** 

264 Photonic Crystals – Introduction, Applications and Theory

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[20] C. J. S. de Matos , J.R. Taylor, K.P. Hansen, "All-fibre Brillouin laser based on holey fibre yielding comb-like spectra," Optics Communications, No.238, pp.185–189, 2004. [21] Z. Yusoff, J. H. Lee, W. Belardi, M. Ibsen, T.M. Monro, and D.J. Richardson, Conference

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**13** 

*France* 

**Overview of Computational Methods** 

A photonic crystal (PC) is a periodic structure whose refraction index of the material is periodically modulated on the wavelength scale to affect the electromagnetic wave propagation by creating photonic band gaps. In 1887, Lord Rayleigh is the first to show a band gap in one-dimensional periodic structures *i.e.* a Bragg mirror. In 1987, Eli Yablonovitch and Sajeev John have extended the band gap concept to the two and threedimensional structures and for the first time, they use the term "photonic crystal"

Progress in computational methods for the photonic crystals is understood through an historical review (Oyhenart, 2005). At the beginning of research in the photonic crystals, the purpose was to find a structure with complete band gap by improving the computational methods. In 1988, John shows theoretically by the scalar method of Korringa-Kohn-Rostoker (KKR) that the face centered cubic lattice (FCC) has a complete band gap between the second and the third band. One year later, Yablonovitch builds this structure and finds a band gap experimentally but the W-point raises a problem. In 1990, Satpathy et al. and Leung et al. confirm the complete band gap by the scalar plane wave method (PWM). A few months later, these two teams improve their methods to obtain vectorial PWM on **D** and **E**  fields. They find that FCC structure does not have complete band gap because W-point and U-point are degenerate. With these results, the editor of the journal "Nature" writes *"Photonic Crystals bite the dust"* (Maddox, 1990). Only two weeks later, Ho et al. created the vectorial PWM on **H** and they do not find the complete band gap in FCC structure but they show a complete band gap in the diamond lattice. In 1992, Sözuer et al. improve convergence of the PWM and they obtain a complete band gap for FCC lattice between 8th and 9th band. This structure that has caused many discussions has a complete band gap but

To study and understand the propagation of the electromagnetic fields in the photonic crystals, computational methods were improved by using their symmetries and periodicities. We will study the classical methods for microwave devices such as the finite element method and the finite difference time domain. After some modifications of these methods, we obtain the band structure of PC which can be calculated by the methods from the solid state physics. For example the plane wave method, the tight binding method and the multiple-scattering theory will be studied. All these computational

**1. Introduction** 

(Yablonovitch, 1987; John, 1987).

not where it was expected.

**for Photonic Crystals** 

Laurent Oyhenart and Valérie Vignéras *IMS Laboratory, CNRS, University of Bordeaux 1* 
