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Problems in Electrostatic Approximation

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Canonical Problems in the Theory of Plasmonics

Part of the book series: Springer Series in Optical Sciences ((SSOS,volume 230))

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Abstract

In this chapter, some electrostatic boundary-value problems involving bounded electron gases with different geometries are studied. By considering the local Drude model , where the dielectric function is calculated by neglecting the spatial nonlocal effects, Laplace’s equation in planar, cylindrical, and spherical geometries is solved, using the separation of variables technique. For brevity, in many sections of this chapter the \(\exp (-i\omega t)\) time factor is suppressed. Furthermore, all media under consideration are nonmagnetic and attention is only confined to the linear phenomena.

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Change history

  • 12 November 2020

    This book was inadvertently published without updating the following corrections.

Notes

  1. 1.

    Here, an insulator means a medium whose relative dielectric constant is assumed to be frequency independent.

  2. 2.

    In the sharp boundary model the thickness of the transition layer of the EG density is much shorter than the wavelength.

  3. 3.

    The method of separation of variables is applied to Laplace’s equation inside and outside the separation surface. This means that the solution for the electrostatic potential Φ can be written as the product of single-variable functions [4].

  4. 4.

    Here, we consider the case, where ε b = 1.

  5. 5.

    The first theoretical description of SPs was presented by Ritchie in 1957 [5].

  6. 6.

    If we reverse the EG slab geometry and study a vacuum gap in an EG, we find the same dispersion relation as for an EG slab, i.e., (2.48). However, we note that in the presence of retardation and/or spatial nonlocal effects they are not.

  7. 7.

    The plasmon hybridization theory (that is a nonretard method) can predict the location of plasmon resonances in complex systems based on knowledge of the resonant behavior of elementary building blocks. It thus provides a powerful tool for optical engineers in the design of functional plasmonic nanostructures. This method was first presented by Prodan et al. [25].

  8. 8.

    The external free charge densities on the separation surfaces of slab are zero.

  9. 9.

    We note by choosing Φ(x, z) as \(-\dfrac {1}{k_{x}}\tilde {\varPhi }(z)\sin k_{x}x\), the results of induced charge density correspond to the nonretarded results in Sect. 3.2.2. However, here the qualitative description of the problem is desired.

  10. 10.

    The polarizability of a circular cylinder has two components, known as axial and transversal polarizabilities. In the present case, i.e., when the length-to-diameter ratio of the system goes to infinity, the axial polarizability is equal to ε 1 − ε 2 [35].

  11. 11.

    This resonance frequency red-shifts as ε 2 is increased.

  12. 12.

    Here, the optical properties are expressed in terms of widths, which are defined as the cross sections per unit length of the cylinder.

  13. 13.

    The sum of absorption and scattering is called extinction.

  14. 14.

    The EG sphere has a uniform polarization P along the z-axis (along the external field); it is equal to \(\mathbf {p}/V=\mathbf {p}\left (4\pi a^{3}/3\right )^{-1} \).

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  1. Kermanshah University of Technology, Kermanshah, Iran

    Afshin Moradi

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Moradi, A. (2020). Problems in Electrostatic Approximation. In: Canonical Problems in the Theory of Plasmonics. Springer Series in Optical Sciences, vol 230. Springer, Cham. https://doi.org/10.1007/978-3-030-43836-4_2

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