Zhurnal Radioelektroniki - Journal of Radio Electronics. eISSN 1684-1719. 2022. №10
ContentsFull text in Russian (pdf)
DOI: https://doi.org/10.30898/1684-1719.2022.10.8
The plane wave expansion method
for investigating the dispersion properties
of metallic photonic crystals
A.Yu. Vetluzhsky
Institute of Physical Materials Science of the Siberian Branch
of the Russian Academy of Sciences
670047, Russia, Ulan-Ude, Sakhyanovoy st., 6
The paper was received June 6, 2022.
Abstract. Original method for studying the dispersion characteristics of photonic crystals – media with a dielectric constant that varies periodically in space – is considered. The method is based on the representation of the wave functions and permittivity of a periodic medium in the form of Fourier series and their subsequent substitution into the wave equation, which leads to the formulation of the dispersion equation. Using the latter, for each value of the wave vector it is possible determined a set of eigen frequencies. Each of eigen frequency forms a separate dispersion curve as a continuous function of the wave number. The Fourier expansion coefficients of the permittivity, which depend on the vectors of the reciprocal lattice of the photonic crystal, are determined on the basis of data on the geometric characteristics of the elements that form the crystal, their electrophysical properties and the density of the crystal. The solution of the dispersion equation found makes it possible to obtain complete information about the number of modes propagating in a periodic structure at different frequencies, and about the possibility of forming band gaps, i.e. frequency ranges within which wave propagation through a photonic crystal is impossible. The focus of this work is on the application of this method to the analysis of the dispersion properties of metallic photonic crystals. The difficulties that arise in this case due to the presence of intrinsic dispersion properties of the metals that form the elements of the crystal are overcome by an analytical description of their permittivity based on the model of free electrons. As a result, a dispersion equation is formulated, the numerical solution of which is easily algorithmized. That makes possible to determine the dispersion characteristics of metallic photonic crystals with arbitrary parameters. Obtained by this method the results of calculation of dispersion diagrams, which characterize two-dimensional metal photonic crystals, are compared with experimental data and numerical results obtained using the method of self-consistent equations. Their good agreement is demonstrated.
Key words: numerical methods, photonic crystals, Brillouin zones, dispersion characteristics, band gaps, spectrum.
Financing: The work was carried out within the framework of the state task of the Ministry of Education and Science of the Russian Federation (subject No. 0270-2021-0004).
Corresponding author: Vetluzhsky Alexander Yurievich, vay@ipms.bscnet.ru
References
1. Yablonovitch E. Inhibited spontaneous emission in solid-state physics and electronics. Phys. Rev. Lett. 1987. V.8. №20. P.2059-2062. https://doi.org/10.1103/PhysRevLett.58.2059
2. Kravchenko V.F., Labun'ko O.S., Lerer A.M., Sinyavskij G.P. Vychislitel'nye metody v sovremennoj radiofizike [Computational methods in modern radiophysics]. Moscow, Fizmatlit Publ. 2009. 469 p. (In Russian)
3. Grinev A.Yu., Gigolo A.I. Matematicheskie osnovy i metody resheniya zadach elektrodinamiki [Mathematical foundations and methods for solving problems of electrodynamics]. Moscow, Radiotekhnika Publ. 2015. 216 p. (In Russian)
4. Gao Y.-J., Yang H.-W., Wang G.-B. A research on the electromagnetic properties of Plasma Photonic Crystal based on the Symplectic Finite-Difference Time-Domain method. Optik. 2016. V.127. №4. P.1838-1841. https://doi.org/10.1016/j.ijleo.2015.11.089
5. Lourtioz J.-M., Benisty H., Berger V., et al. Photonic Crystals: Towards Nanoscale Photonic Devices. 2nd edition. Springer. 2008. 513 р.
6. Vetluzhsky A.Yu. Waveguides based on linear defects in metal electromagnetic crystals. Technical Physics. 2017. V.62. №1. С.178-182. https://doi.org/10.1134/S106378421701025X
7. Bejtmen G., Erdeji A. Funkcii Besselya, funkcii parabolicheskogo cilindra, ortogonal'nye mnogochleny. Vysshie transcendentnye funkcii [Bessel functions, parabolic cylinder functions, orthogonal polynomials. Higher transcendental functions]. Vol. 2. Moscow, Nauka Publ. 1974. 296 p. (In Russian)
8. McGurn A.R., Maradudin A.A. Photonic band structures of two- and three-dimensional periodic metal or semiconductor arrays. Phys. Rev. B. 1993. V.48. №23. P.17576-17579. https://doi.org/10.1103/PhysRevB.48.17576
9. El-Kady I., Sigalas M.M., Biswas R., Ho K.M., Soukoulis C.M. Metallic photonic crystals at optical wavelengths. Phys. Rev. B. 2000. V.62. №23. P.15299-15302. https://doi.org/10.1103/PhysRevB.62.15299
10. Vetluzhsky A.Yu. Effective electrophysical properties of metallic electromagnetic crystals. Zhurnal radioelectroniki [Journal of Radio Electronics] [online]. 2015. №1. (In Russian)
11. Vetluzhsky A.Yu. Method of self-consistent equations in solving problems of wave scattering on systems of cylindrical bodies. Komp'yuternye issledovaniya i modelirovanie [Computer Research and Modeling]. 2021. V.13. №.4. P.725-733. https://doi.org/10.20537/2076-7633-2021-13-4-725-733 (In Russian)
12. Kittel C. Introduction to Solid State Physics (4th ed.). New York, John Wiley & Sons Publ. 1971.792 p.
For citation:
Vetluzhsky A.Yu. The plane wave expansion method for investigating the dispersion properties of metallic photonic crystals. Zhurnal radioelektroniki [Journal of Radio Electronics] [online]. 2022. №10. https://doi.org/10.30898/1684-1719.2022.10.8 (In Russian)