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Fundamentals of Plasmonics

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From Classical to Quantum Plasmonics in Three and Two Dimensions

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Abstract

We first review the necessary components of electrodynamics and establish a practical notion of optical excitations. Next, we specialize to plasmonic excitations, and discuss their classical features. Finally, we cover theoretical aspects of techniques that probe the properties of plasmons in practice.

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Notes

  1. 1.

    Equations (2.1) collect the historical Faraday law (a), the Maxwell–Ampère law (b), and the electric and magnetic Gauss laws (c) and (d).

  2. 2.

    The Maxwell equations can in fact be written in just three equations, namely Eqs. (2.1a), (2.1b), (2.2) from which Eqs. (2.1c) and (2.1d) follow [2].

  3. 3.

    Anisotropy can be included in Eqs. (2.3) by considering tensorial forms of the response functions \(\varepsilon _{\textsc {b}}\) and \(\sigma \).

  4. 4.

    We define Fourier transform pairs in time and space:

    $$\begin{aligned} \begin{aligned} f(\omega )&\equiv \int f(t)\mathrm {e}^{ \mathrm {i} \omega t} \, \mathrm {d} {t},\\ g( \mathbf {k} )&\equiv \int g( \mathbf {r} )\mathrm {e}^{- \mathrm {i} \mathbf {k} \cdot \mathbf {r} } \, \mathrm {d} { \mathbf {r} }, \end{aligned} \qquad \qquad \qquad \qquad \begin{aligned} f(t)&\equiv \frac{1}{2\pi }\int f(\omega )\mathrm {e}^{- \mathrm {i} \omega t} \, \mathrm {d} {\omega },\\ g( \mathbf {r} )&\equiv \frac{1}{(2\pi )^3}\int g( \mathbf {k} )\mathrm {e}^{ \mathrm {i} \mathbf {k} \cdot \mathbf {r} } \, \mathrm {d} { \mathbf {k} }. \end{aligned} \end{aligned}$$
  5. 5.

    Equivalently, in the time-domain: \(\partial _t \tilde{ \mathbf {D} } \equiv \partial _t \mathbf {D} + \mathbf {J} _{\scriptscriptstyle \mathrm {ind}} \).

  6. 6.

    The optical excitations deduced from Eq. (2.7b) are not generally guaranteed to be bound excitations; such a distinction can be made by using an optical potential \(\Delta \varepsilon ( \mathbf {r} )\) relative to the background [3].

  7. 7.

    The equivalent real-space definition reading .

  8. 8.

    In the presence of material loss, or indeed even of radiation loss, the optical excitation frequencies are complex; the imaginary part provides the excitation’s inverse life-time.

  9. 9.

    Here assuming full isotropy and correspondingly scalar \(\varepsilon ( \mathbf {r} , \mathbf {r} ';\omega )\).

  10. 10.

    The specification is occasionally made in terms of the Wigner-Seitz radius \(r_s\) defined as the radius of a spherical volume containing on average one electron \(\tfrac{4\pi }{3} r_s^3 = 1/n_{0}\) [9].

  11. 11.

    When combined with a Drude model, such a model is typically referred to as a Lorentz–Drude model.

  12. 12.

    Not just metals but also strongly doped semiconductors are well described by a Drude model [17, 18]—even graphene, a semi-metal with nonparabolic dispersion, exhibits a (modified) Drude response, as we return to in Part II.

  13. 13.

    The full dispersion is obtained from the zeros of the Lindhard dielectric function \(\varepsilon _{\textsc {l}}(k,\omega ) = 1-V(k)\chi _0(k,\omega )\), with Coulomb interaction \(V(k)=e^2/\varepsilon _0 k^2\) and noninteracting density-density response \(\chi _0(k,\omega )\) given by products of fractions and logarithms of k [9, 13].

  14. 14.

    The TE and TM polarization are commonly also referenced as s- and p-polarization, respectively, indicating parallel and senkrecht (German; perpendicular) polarization relative to the plane of incidence.

  15. 15.

    The corresponding TE reflection coefficient is \(r_{\textsc {te}} = ( k_{\scriptscriptstyle \perp } ^+ - k_{\scriptscriptstyle \perp } ^-)/( k_{\scriptscriptstyle \perp } ^+ + k_{\scriptscriptstyle \perp } ^-)\).

  16. 16.

    The polariton-epithet indicates the mixing with the free polariton field in the dielectric medium, i.e. the inclusion of retardation. In the nonretarded limit it is omitted; the \( k_{\scriptscriptstyle \parallel } /k_0\rightarrow \infty \) solution of Eq. (2.17) is thus simply denoted the surface plasmon (SP) [23].

  17. 17.

    Back-bending is absent if \(\omega \) rather than \( k_{\scriptscriptstyle \parallel } \) is chosen complex [24, 25], in this case, the relevant decay involves times rather than lengths.

  18. 18.

    Equation (2.18) may be obtained by solving the Poisson equation \(\nabla \cdot [\varepsilon ( \mathbf {r} )\nabla \phi ( \mathbf {r} )] = 0\) for the potential \(\phi ( \mathbf {r} )\) via expansion in multipole solutions of the Laplace equation, matched so as to ensure continuity at the boundary.

  19. 19.

    The general sum rule, not restricted to specific choices of the dielectric or bound response, is \(\Delta (\omega _{-}) + \Delta (\omega _{+}) = 1\) with \(\Delta (\omega ) \equiv {\varepsilon ^{\scriptscriptstyle \text {d}} }/[\varepsilon ^{\scriptscriptstyle \text {d}} - \varepsilon ^{\scriptscriptstyle \text {m}} ( \omega )]\) for dielectric functions \(\varepsilon ^{\scriptscriptstyle \text {d}} \) and \(\varepsilon ^{\scriptscriptstyle \text {m}} (\omega )\) corresponding to the dielectric and metal regions, respectively [31]. This can readily be derived from the LRA-limit of Eq. (3.38a).

  20. 20.

    Unsurprisingly, the single-channel limit is ubiquitous also outside plasmonics, e.g. in atomic physics in the interaction between classical radiation and two-level systems [1].

  21. 21.

    As first noted by Rayleigh [56], the scaling \(\sigma _{\scriptscriptstyle \text {sca}} \sim \lambda ^{-4}\) qualitatively accounts for the color of the sky: light scattering off airborne molecules is strongest at short wavelengths, thus entailing a blue appearance.

  22. 22.

    A full discussion of the optical LDOS requires consideration of both electric and magnetic contributions [70]. The former, however, is arguably more important than the latter, at least in a nanophotonic context, since it relates directly with the properties of electric dipoles and thus real emitters.

  23. 23.

    The operator form of the dyadic Green function was noted in Sect. 2.1.2. Its real-space form is obtained by taking appropriate matrix elements , such that .

  24. 24.

    Decay enhancement (or quenching) is a cornerstone of cavity quantum electrodynamics, wherein it is known as the Purcell factor.

  25. 25.

    In evaluating Eq. (2.24) for practical systems it is often advantageous to make use of the following classical relation: \(\frac{\rho _{\hat{\mathbf {n}}}^{\textsc {e}}( \mathbf {r} ,\omega _{\scriptscriptstyle 12} )}{\rho _{\scriptscriptstyle 0} ^{\textsc {e}}(\omega _{\scriptscriptstyle 12} )} = \frac{6\pi \varepsilon _{\scriptscriptstyle 0} \text {Im}[\mathbf {p}_{\scriptscriptstyle \text {c}} ^* \cdot \mathbf {E} ( \mathbf {r} )]}{|\mathbf {p}_{\scriptscriptstyle \text {c}} |^2 k_0^3}\) with \( \mathbf {E} \) denoting the total field due to a classical dipole \(\mathbf {p}_{\scriptscriptstyle \text {c}} \) at point \( \mathbf {r} \), with evaluation at frequency \(\omega _{\scriptscriptstyle 12} \) understood [11].

  26. 26.

    Implicit in Eq. (2.26) is the assumption that the electron’s path \( \mathbf {r} _{\scriptscriptstyle \text {e}} (t)\) is not appreciably modified by the interaction with the induced field, i.e. an assumption of linear response, justified by the enormous differences in total (hundreds of keV) and lost (few eV) energy; this is sometimes termed the no-recoil approximation.

  27. 27.

    The identification with a generalized LDOS requires replacing by , with vacuum contribution . The replacement makes no change to \(\Gamma (\omega )\) because the contribution vanishes when integrated, reflecting the fact that the electron does not incur loss on itself in vacuum.

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    Thomas Christensen

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Christensen, T. (2017). Fundamentals of Plasmonics. 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_2

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