Singular perturbations approach to localized surface-plasmon resonance: Nearly touching metal nanospheres

Metallic nano-structures characterised by multiple geometric length scales support low-frequency surface-plasmon modes, which enable strong light localization and field enhancement. We suggest studying such configurations using singular perturbation methods, and demonstrate the efficacy of this approach by considering, in the quasi-static limit, a pair of nearly touching metallic nano-spheres subjected to an incident electromagnetic wave polarized with the electric field along the line of sphere centers. Rather than attempting an exact analytical solution, we construct the pertinent (longitudinal) eigen-modes by matching relatively simple asymptotic expansions valid in overlapping spatial domains. We thereby arrive at an effective boundary eigenvalue problem in a half-space representing the metal region in the vicinity of the gap. Coupling with the gap field gives rise to a mixed-type boundary condition with varying coefficients, whereas coupling with the particle-scale field enters through an integral eigenvalue selection rule involving the electrostatic capacitance of the configuration. By solving the reduced problem we obtain accurate closed-form expressions for the resonance values of the metal dielectric function. Furthermore, together with an energy-like integral relation, the latter eigen-solutions yield also closed-form approximations for the induced-dipole moment and gap-field enhancement under resonance. We demonstrate agreement between the asymptotic formulas and a semi-numerical computation. The analysis, underpinned by asymptotic scaling arguments, elucidates how metal polarization together with geometrical confinement enables a strong plasmon-frequency redshift and amplified near-field at resonance.Comment: 13 pages, 7 figure

Asymptotic network models of subwavelength metamaterials formed by closely packed photonic and phononic crystals

We demonstrate that photonic and phononic crystals consisting of closely spaced inclusions constitute a versatile class of subwavelength metamaterials. Intuitively, the voids and narrow gaps that characterise the crystal form an interconnected network of Helmholtz-like resonators. We use this intuition to argue that these continuous photonic (phononic) crystals are in fact asymptotically equivalent, at low frequencies, to discrete capacitor-inductor (mass-spring) networks whose lumped parameters we derive explicitly. The crystals are tantamount to metamaterials as their entire acoustic branch, or branches when the discrete analogue is polyatomic, is squeezed into a subwavelength regime where the ratio of wavelength to period scales like the ratio of period to gap width raised to the power 1/4; at yet larger wavelengths we accordingly find a comparably large effective refractive index. The fully analytical dispersion relations predicted by the discrete models yield dispersion curves that agree with those from finite-element simulations of the continuous crystals. The insight gained from the network approach is used to show that, surprisingly, the continuum created by a closely packed hexagonal lattice of cylinders is represented by a discrete honeycomb lattice. The analogy is utilised to show that the hexagonal continuum lattice has a Dirac-point degeneracy that is lifted in a controlled manner by specifying the area of a symmetry-breaking defect

Surface-plasmon resonances of arbitrarily shaped nanometallic structures in the small-screening-length limit

According to the hydrodynamic Drude model, surface-plasmon resonances of metallic nanostructures blueshift owing to the nonlocal response of the metal's electron gas. The screening length characterising the nonlocal effect is often small relative to the overall dimensions of the metallic structure, which enables us to derive a coarse-grained nonlocal description using matched asymptotic expansions; a perturbation theory for the blueshifts of arbitrary shaped nanometallic structures is then developed. The effect of nonlocality is not always a perturbation and we present a detailed analysis of the "bonding" modes of a dimer of nearly touching nanowires where the leading-order eigenfrequencies and eigenmode distributions are shown to be a renormalisation of those predicted assuming a local metal permittivity

Asymptotics of surface-plasmon redshift saturation at sub-nanometric separations

Many promising nanophotonics endeavours hinge upon the unique plasmonic properties of nanometallic structures with narrow non-metallic gaps, which support super-concentrated bonding modes that singularly redshift with decreasing separations. In this letter, we present a descriptive physical picture, complemented by elementary asymptotic formulae, of a nonlocal mechanism for plasmon-redshift saturation at subnanometric gap widths. Thus, by considering the electron-charge and field distributions in the close vicinity of the metal-vacuum interface, we show that nonlocality is asymptotically manifested as an effective potential discontinuity. For bonding modes in the near-contact limit, the latter discontinuity is shown to be effectively equivalent to a widening of the gap. As a consequence, the resonance-frequency near-contact asymptotics are a renormalisation of the corresponding local ones. Specifically, the renormalisation furnishes an asymptotic plasmon-frequency lower bound that scales with the $1/4$-power of the Fermi wavelength. We demonstrate these remarkable features in the prototypical cases of nanowire and nanosphere dimers, showing agreement between our elementary expressions and previously reported numerical computations