Determination of the Allowed Frequencies of Electromagnetic Wave Propagation in 2D Photonic Crystal by the Coupled Waves Method (CWM)

Abstract

The paper demonstrates that at imposition of periodic boundary conditions and using the coupled waves method (CWM) which is applied previously for the analysis of diffraction of electromagnetic waves in plane 1D, grating it is possible to quickly define if the given frequency of the electromagnetic wave is allowed for propagation in a 2D photonic crystal. The problem is reduced to the problem of higher algebra on the eigen values and the eigen vectors of the kind  and to verification if the module of  is equal to one. If , the given frequency is allowed. At a calculation the dimension of vector  is equal to the doubled number of the used coupled waves  and is determined by the required accuracy of analysis. High accuracy of determination of the allowed frequencies of branches of band structure is secured at, which determines the band gap of the photonic crystal. Due to the fact that standard photonic crystals possess symmetrical spatial dependence of the dielectric constant the use of symmetry depending upon its type permits to pass on to the dimension of  vector  or  without the loss in accuracy of the analysis with simultaneous diminishing of calculation time by approximately 8 times.

References

1. Karachevtseva, I.A., (2004), Two-dimensional photonic crystals as perspective materials of modern nanoelectronics, Semiconductor Physics, Quantum Electronics & Optoelectronics. 7(4):430-435.
2. Yablonovitch, E., (1997), Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett. 58(17):2059-2065.
3. Yablonovitch, E., Gmitter, T.J., and Leung, K.M., (1991), Photonic band structure: the face-centered-cubic case employing non spherical atoms, Phys. Rev. Lett. 67(17):2295-2298.
4. Joannopoulis, J.D., Meade, R.D., and Winn, J.N., (1995), Photonic Crystals: Molding the Flow of Light. Princeton: University Press. - 137 p.
5. Foteinopoulou, S., and Soukoulis, C.M., (2002), Negative refraction and left-hand behavior in two-dimensional photonic crystals, arXiv:cond-mat 0212434. -5 p.
6. Xu, U., Wang, J., He, Q., Cao, L., Su, P., and Jin, G., (2005), Optical filter based on contra-directional waveguide coupling in 2D photonic crystal with square lattice of dielectric rods, Optics Express. 13(15):5608-5613.
7. Khyzhnyak, N.А., (1957), Artificial anisotropy of dielectrics, Journ. of Techn.Physics. 27(9):2006-2033 (in Russian).
8. Skyrta, Ye.А., and Khyzhnyak, N.А., (1982), Dispersion properties of artificial anisotropic dielectrics. Kharkiv: Publication of IRE AS of UkrSSR, 39 p. (in Russian).
9. Khyzhnyak, N.А., (1986), Integral equations of macroscopic electrodynamics, Naukova dumka, Kiev: 280 p. (in Russian).
10. Seydon, F., Ramahi, O.M., Duraiswami, R., and Seppanen, T., (2006), Numerical computation of the Green’s function for two-dimensional finite-size photonic crystals of infinite length, Optics Express. 14(23):11362-11371.
11. Dansas, P., and Paraire, N., (1998), Fast modelling of photonic band gap structures by use of diffraction grating approach, J. Opt. Soc. Am. A. 15(6):1586-1597.
12. Glushko, A.E., Glushko, E.Ya., and Karachevtseva, I.A., (2005), Theory of two-dimensional photonic crystals with lamellar cylindrical pores, Semiconductor Physics, Quantum Electronics & Optoelectronics. 8(1):64-71.
13. Sözüer, H.S., Haus, J.W., and Inguva, R., (1992), Photonic bands: Convergence problems with the plane-wave method, Phys. Rev. B. 45(24):13962-13073.
14. Fityo, V.М., Bobitski, Ya.V., and Laba, H.P., (2005), Analysis of band structure of 1D and 2D photonic crystals using the coupled waves method, Radio Physics and Electronics. 10(1):123-131 (in Russian).
15. Moharam, M.G., Grann, E.B., Pommet, D.A., and Gaylord, T.K., (1995), Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary grating, J. Opt. Soc. Am. A. 12(5):1068-1076.
16. Luan, Pi-G., and Ye, Zh., (2001), Two dimensional photonic crystals, ArXiv: cond-mat/0105428. - 15 p.
17. Fityo, V.М., and Bobitski, Ya.V., (2004), Аnalysis of planar gratings with symmetrical properties (CWM), Procs. of National University „Lvivska Polytekhnika”, Electronics. 513:203-213 (in Russian).
18. Fityo, V.M., and Bobitski, Ya.V., (2005). Diffraction analysis by periodic structures using a method of coupled waves, Opto-Electronics Review. 13(4):P.331-339.
19. Li, L., (1995), Use of Fourier series in the analysis of discontinuous periodic structures,
    J. Opt. Soc. Am.A. 12(5):1870-1876.
20. Li, L., (2003), Note on the S-matrix propagation algorithm, J. Opt. Soc. Am. A. 20(4):655-660.

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