**12. Conclusion: Comparison of computational method**

The computational methods are studied and compared on table 1. The first four methods are three-dimensional numerical methods coming from electromagnetism. The FEM gives very precise results but it requires many resources systems. Another method, the FDTD is very used and converges without an excessive mesh thanks to its formulation. If you want more precise results, the mesh becomes heavy and you need to use FEM. The FDFD studies photonic crystals with a high number of layers by modeling only one layer. The FIT is a method which studies numerically PC with a high number of objects, without holding excessive resources system but the result is approximate. These methods calculate dielectric and metallic PC to obtain reflection and transmission coefficients and the band structure.

Then, we have others methods resulting from the solid state physics which require a very small computing time. They were adapted from the scalar methods of the solid state physics. The PWM calculate only the band structure. The tight binding method is less used than the PWM although this method is fast for the calculation of defect states in a PC. The multiple-scattering theory which is also used in optics studies analytically large finite PC and it easily takes the defects into account in the PC. For three-dimensional PC, there is no simple method, fast and accurate. We must remove a feature. For example, the 3D-MST is fast and accurate but the computer program is complex to write.


Abbreviations:

Q.M. : quantum mechanics and solid state physics

E.M. : electromagnetism

Freq. : frequency domain Time : time domain

Analyt. : analytical method Num. : numerical method Med. : medium Cyl. : cylinder

Table 1. Specifications of the computational methods

**0**

**14**

*Poland*

**Coupled Mode Theory of Photonic Crystal Lasers**

Photonic crystals (PC) are structures with periodic variation of the refractive index in one, two or three spatial dimensions. The dynamic development of experimental and theoretical work on photonic crystals has been launched by Yablonovitch (1987; 1993) and Sajeev John (1987) publications, although the idea of periodic structures had been known since Strutt (1887).

The main property of photonic crystal is the existence of a frequency range, for which the propagation of electromagnetic waves in the medium is not permitted. These frequency ranges are commonly known as photonic band gaps, giving the ability to modify the structure parameters, e.g. group velocity, coherence length, gain, and spontaneous emission. This type

Much of the research on active structures is devoted to efficient photonic sources of coherent radiation. Photonic crystals are one of these structures, and they are used in lasers as mirrors (Dunbar et al. (2005); Scherer et al. (2005)), active waveguides (Watanabe & Baba (2006)), coupled cavities (Steinberg & Boag (2006)), defect microcavities (Asano et al. (2006); Lee et al.

Lasers with defect two-dimensional photonic crystals are known for their high finesse (Monat

Photonic crystal band-edge lasers allow to obtain edge (Cojocaru et al. (2005)) and surface emission (Turnbull et al. (2003); Vurgaftman & Meyer (2003)) of coherent light from large cavity area. They also allow to control the output beam pattern by manipulation of the primitive cell geometry (Iwahashi et al. (2010); Miyai et al. (2006)), provide low threshold (Susa (2001)), and beams which can be focused to a size less than the wavelength (Matsubara

The photonic crystal structures lasing wavelengths span from terahertz (Chassagneux et al. (2009); Sirigu et al. (2008)), through infrared (Kim et al. (2006)) to visible (Lu et al. (2008);

of periodic structures is used in both passive and active devices.

(2004)), and the laser active region (Cojocaru et al. (2005)).

et al. (2001)) and very low threshold (Nomura et al. (2008)).

**1.1 Two-dimensional photonic crystal lasers**

**1. Introduction**

et al. (2008)).

Zhang et al. (2006)).

Marcin Koba1 and Pawel Szczepanski2

<sup>1</sup>*University of Warsaw, National Institute of Telecommunications*

<sup>2</sup>*Warsaw University of Technology, National Institute of Telecommunications*

### **13. References**

