29 research outputs found
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
Spoof surface plasmons guided by narrow grooves
An approximate description of surface waves propagating along periodically
grooved surfaces is intuitively developed in the limit where the grooves are
narrow relative to the period. Considering acoustic and electromagnetic waves
guided by rigid and perfectly conducting gratings, respectively, the wave field
is obtained by interrelating elementary approximations obtained in three
overlapping spatial domains. Specifically, above the grating and on the scale
of the period the grooves are effectively reduced to point resonators
characterised by their dimensions as well as the geometry of their apertures.
Along with this descriptive physical picture emerges an analytical dispersion
relation, which agrees remarkably well with exact calculations and improves on
preceding approximations. Scalings and explicit formulae are obtained by
simplifying the theory in three distinguished propagation regimes, namely where
the Bloch wavenumber is respectively smaller than, close to, or larger than
that corresponding to a groove resonance. Of particular interest is the latter
regime where the field within the grooves is resonantly enhanced and the field
above the grating is maximally localised, attenuating on a length scale
comparable with the period
Weakly nonlinear dynamics of a chemically active particle near the threshold for spontaneous motion. Part 1: Adjoint method
In this Series, we study the weakly nonlinear dynamics of chemically active
particles near the threshold for spontaneous motion. In this Part, we focus on
steady solutions and develop an 'adjoint method' for deriving the nonlinear
amplitude equation governing the particle's velocity, first assuming the
canonical model in the literature of an isotropic chemically active particle
and then considering general perturbations about that model. As in previous
works, the amplitude equation is obtained from a solvability condition on the
inhomogeneous problem at second order of a particle-scale weakly nonlinear
expansion, the formulation of that problem involving asymptotic matching with a
leading-order solution in a remote region where advection and diffusion are
balanced. We develop a generalised solvability condition based on a Fredholm
Alternative argument, which entails identifying the adjoint linear operator at
the threshold and calculating its kernel. This circumvents the apparent need in
earlier theories to solve the second-order inhomogeneous problem, resulting in
considerable simplification and adding insight by making it possible to treat a
wide range of perturbation scenarios on a common basis. To illustrate our
approach, we derive and solve amplitude equations for a number of perturbation
scenarios (external force and torque fields, non-uniform surface properties,
first-order surface kinetics and bulk absorption), demonstrating that
sufficiently near the threshold weak perturbations can appreciably modify and
enrich the landscape of steady solutions
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 -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
Spontaneous locomotion of a symmetric squirmer
The squirmer is a popular model to analyse the fluid mechanics of a
self-propelled object, such as a micro-organism. We demonstrate that some
fore-aft symmetric squirmers can spontaneously self-propel above a critical
Reynolds number. Specifically, we numerically study the effects of inertia on
axisymmetric squirmers characterised by a 'quadrupolar' fore-aft symmetric
distribution of surface-slip velocity; under creeping-flow conditions, such
squirmers generate a pure stresslet flow, the stresslet sign classifying the
squirmer as either a 'pusher' or 'puller.' Assuming axial symmetry, and over
the examined range of the Reynolds number (defined based upon the
magnitude of the quadrupolar squirming), we find that spontaneous symmetry
breaking occurs in the puller case above , with steady
swimming emerging at the threshold via a supercritical pitchfork bifurcation,
beyond which the swimming speed grows monotonically with