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Introduction

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From Classical to Quantum Plasmonics in Three and Two Dimensions

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Abstract

Plasmons are collective excitations in the free-electron plasma of conductive materials, such as e.g. noble metals. Classically, they are characterized by an oscillation of the free-electron plasma, sustained by inter-electronic interaction in a positive ionic background. The interaction is mediated in the nonretarded case by the Coulomb force, and in a full electrodynamic view, by the Maxwell equations. It is a testament to their remarkable potential, that their unique attributes have been exploited in numerous cases long before their theoretical description.

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Notes

  1. 1.

    For a while, following the 1929 classical description by Tonks and Langmuir [3], plasmons were known as Langmuir waves.

  2. 2.

    The 1998 observations of enhanced transmission through subwavelength hole-arrays [20], constitutes an oft-noted example of such a galvanizing and stimulating event.

  3. 3.

    The intersection of these extensions, i.e. simultaneous accounting of optical and electronic quantization, poses a formidable challenge, which, however, has seen progress recently [27, 28].

  4. 4.

    For brevity, we do not explicitly indicate the magnitude of \(\mathbf {u}( \mathbf {r} ,t)\) because it drops out in the final analysis—implicitly, however, we assume it to be sufficiently small so as to justify a perturbative analysis.

  5. 5.

    We emphasize that it is here, at the accounting of the field induced by the electrons’ displacement, that the Coulomb interaction is included. That it is the Poisson equation which is invoked is only a semantic change; it’s equivalence to a Coulomb interaction \(e^2/| \mathbf {r} - \mathbf {r} '|\) is readily discerned by considering its solution in the case of a single charge \(\delta \rho ( \mathbf {r} ) = -e\delta ( \mathbf {r} )\).

  6. 6.

    Using the integral identities \(\int _{-\infty }^\infty \mathrm {e}^{ \mathrm {i} kx}(x^2+y^2+z^2)^{-1/2} \, \mathrm {d} x = 2K_0[k(y^2+z^2)^{1/2}]\) and \(\int _0^\infty K_0[k(y^2+z^2)^{1/2}] \, \mathrm {d} {y} = \pi \mathrm {e}^{-k|z|}/2k\) [52] or the differential identity \(\partial _z^2\mathrm {e}^{-k|z|} = -2k\delta (z) +k^2\mathrm {e}^{-k|z|}\) in the Coulomb or Poisson case, respectively.

  7. 7.

    Of course, a basic sort of size- and momentum-dependence has been omitted: that due to retardation, which produces such a dependence in the large-scale and low-momentum regimes; our present efforts, however, are concerned with the opposite limits (and is restricted thereto as well, owing to its quasistatic outlook).

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  1. Department of Physics, Massachusetts Institute of Technology, Cambridge, USA

    Thomas Christensen

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Christensen, T. (2017). Introduction. In: From Classical to Quantum Plasmonics in Three and Two Dimensions. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-48562-1_1

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