Abstract
We study the photonic band structure of a metal–dielectric periodic structure. The metallic component is described by the Drude model; therefore, the electric permittivity is frequency dependent, i.e., dispersive. Rather than solving a nonlinear eigenvalue problem for the band structure of the material, we follow a time-dependent formulation described in Raman and Fan (Phys Rev Lett 104:087401, 2010) which leads to a linear eigenvalue problem. At issue is the question of completeness of the eigenfunctions, which is claimed but not proven in Raman and Fan (2010). We establish completeness in one dimension. We further describe the existence of accumulation points in the spectrum that lead to an infinite family of ‘zero group velocity’ waves. Numerical calculations illustrate some of the main ideas of this work.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 1, pp. 358–361. Interscience Publishers, Geneva (1953)
El-Kady, I., Sigalas, M., Biswas, R., Ho, K., Soukoulis, C.: Metallic photonic crystals at optical wavelengths. Phys. Rev. B 62, 15299–15302 (2000)
Hao, X., Sun, J., Zettl, A.: The spectrum of differential operators and square integrable soutions. J. Funct. Anal. 262, 1630–1644 (2012)
Hislop, P., Sigal, I.: Introduction to Spectral Theory with Application to Schrödinger’s Equation. Springer, Berlin (1996)
Raman, A., Fan, S.: Photonic band structure of dispersive metamaterials formulated as a Hermitian eigenvalue problem. Phys. Rev. Lett. 104, 087401 (2010)
Reed, M., Simon, B.: Functional Analysis, Volume I: Methods of Modern Mathematical Physics, Revised and enlarged edition, Academic Press, INC., Cambridge (1980)
Scalora, M., Bloemer, M., Pethel, A., Dowling, J., Bowden, C., Manka, A.: Transparent, metallo–dielectric, one-dimensional, photonic band-gap structures. J. Appl. Phys. 83, 2377–2383 (1998)
Toader, O., John, S.: Photonic band gap enhancement in frequency-dependent dielectrics. Phys. Rev. E 70, 046605 (2004)
Wu, C., Chung, Y., Syu, B., Yang, T.: Band gap extension in a one-dimensional ternary metal–dielectric photonic crystal. Prog. Electromagn. Res. PIER 102, 81–93 (2010)
Yee, K.: Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propag. 14, 302–307 (1966)
Acknowledgements
Many people have contributed to this work through discussions. These include David Dobson, Michael Weinstein, Junshan Lin, Stephen Shipman, Braxton Osting, and Eric Bonnetier. This project was initiated at Hong Kong University of Science and Technology where FS was a visiting professor in January 2019. He thanks his hosts for the hospitality during the productive stay. The research of FS is funded in part by the National Science Foundation under DMS 1440471. The research of HZ is partially funded by Hong Kong RGC Grant GRF 16306318 and GRF 16304517.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Santosa, F., Zhang, H. Photonic band gap phenomenon in a metal–dielectric periodic structure. Res Math Sci 7, 15 (2020). https://doi.org/10.1007/s40687-020-00211-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40687-020-00211-w