Quantum Physics-Based Modelling of Frequency-Dependent Dielectric Function for Small-Scale Devices in Optical Communications Lateef Adesola Akinyemi, Student Member, IEEE * and Alireza Baghai-Wadji, Senior Member, IEEE † Department of Electrical Engineering, University of Cape Town, South Africa Email: * ltfaki001@myuct.ac.za, † alireza.baghai-wadji@uct.ac.za Abstract—This paper describes the one dimensional quantum physics-based modelling of frequency-dependent dielectric func- tion for small-scale devices. The eigenpairs of the underlying canonical and associated perturbed quantum systems are com- puted and utilized for the calculation of the dielectric function. Galerkin method has been employed to discretize the boundary value problem of interest. The starting point is the infinite quantum potential well problem, for which the complete set of eigenfunctions is readily available in closed-form. Subsequently, the wavefunction of the perturbed system is expressed as a linear combination of the original eigenfunctions and the eigenpairs determined. Two- and three dimensional problems can be solved mutatis mutandis. The introduction of sinc functions throughout the analysis ensures the robustness of the computations. The analytical and numerical results demonstrate that the real- and imaginary parts of the dielectric function are even and odd functions, respectively, as expected. Index Terms—Dielectric function, frequency-dependence, quantum physics, small-scale devices. I. I NTRODUCTION Progress in the field of nanophotonics and plasmonics has been rapid in recent times. Consequently, the metallic spherical semi-shell of nano and micro sizes have received consid- erable attention due to their anisotropic, optical, and high tunability features such as light confinement, light focusing, field enhancement, localization of fields near the edges, and absorption increase at long wavelengths in comparison with their spherical structures. Therefore, these characteristics and properties of the structure are anticipated to have ability and potential applications in various fields of engineering such as transistors, lasers, quantum computers, quantum communica- tion systems, and nano-antennas [1]–[9]. The metallic geometry for the problem in this study is a thin layered material. The properties of the structure depend not only on its material properties such as dielectric, permeability, susceptibility, and refractive index but also geometric form such as size, shape and so on. When the size of a metallic structure is large (bulk materials), the material properties (permittivity, permeability, susceptibility, and refractive index) do not depend on the geometrical properties (shape and size). However, if the dimensions of the structure are considerably less than the mean free path electron, then the material properties will depend on the geometrical properties as a result of confinement effects of the electrons. Owing to the reducing scale of devices as optical communication system technology advances, the current models for the frequency-dependent dielectric function needs to be remodelled to operate optimally in the nano region. This remodelling is the objective of this paper. The characteristics of the method has been shown in terms of the simplest possible canonical problem. Extension to higher dimensions is immediate. The key contributions of the paper are: • Sinc functions have been introduced through the discus- sion to ensure numerical robustness and well-posedness of the calculations. • Closed-form expressions have been derived for the di- electric function and matrix element for the nanowire. • The dielectric function model for a 1-D silver nanowire has been computed numerically and the results are pre- sented. The paper is organised as follows.The related work is reviewed in Section II. Section III discusses the underlying theory, method and derives the dielectric function model. Section IV presents and discusses the numerical results. Finally, Section V concludes the paper. II. RELATED WORK With further reduction in the dimensions of the metallic spherical structure until they are comparable to the wavelength of the particle (electron) wave function, certain phenomena such as quantum effects will be exhibited. Classically, these effects cannot be explained unless the quantum mechanical ap- proach is applied. The quantum size effects in metal particles and thin films using an extended random phase approximation (RPA) were presented in [1]–[3]. This theory is applied only to a simple free electron gas model for small particles, thus ana- lytical expressions are obtained for electronic polarizability of a thin metallic film for optical properties in plasma resonance regime. In [4], the excitons in inhomogeneous quantum dots was investigated in a strong confinement region. These dots possess an internal structure, with an inner core that acts like a repulsive potential for the particles using a macroscopic model for the analysis of the dielectric discrepancy at the boundaries. Besides, in [5], the contributions to the plasmon line shape of metallic nano shells were probed in the range of 100-250 nm 78 978-1-5386-1098-5/17/$31.00 '2017 IEEE