Biomedical Engineering Reference
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of Fig. 11.6 (i.e., increasing numerical values from blue to red
colors). The non-uniform color distributions shown in Figs. 11.7
graphically represent the distinctive spatial order of Vogel's spirals.
In particular, we clearly notice that circular symmetry is found
in the distribution of particles for all the spirals, including the
strongly inhomogeneous
-series. As recently demonstrated
by Trevino et al. [37], regions of markedly different values of
d
define “radial heterostructures” that can e
ciently trap radiation
in regions with different lattice constants, similarly to the case of
the concentric rings of omniguide Bragg fibers. The sharp contrast
between adjacent rings radially traps radiation by Bragg scattering
along different circular loops. The circular regions evidenced in
the spatial map of local particle coordination in Fig. 11.7 well
correspond to the scattering rings observed in the Fourier spectra
(Fig. 11.4), and are also at the origin of the recently discovered
circular scattering resonances carrying OAM in Vogel spirals [34,
52]. Moreover, as discussed in Refs. [37, 52], the characteristic
“circular Bragg scattering” occurring between dielectric rods in
Vogel spirals gives rise to localized resonant modes with well-
defined radial and azimuthal numbers similar to the whispering
gallery modes of micro-disk resonators.
We observe that characteristic radial heterostructures are
present in the Vogel spiral geometry regardless of the particular
choice of the irrational divergence angle. In Fig. 11.8, we make
this point by displaying structural results obtained on the
τ
,
μ
,
and
π
-irrational spiral arrays, which vastly differ in their real-
space structures (Fig. 11.2b-d). This analysis unveils very general
structuralfeaturesofaperiodicVogelspiralsgeneratedbyirrational
divergence angles and can guide the design of photonic-plasmonic
structures that support a large density of distinctively localized
optical resonances with a large degree of azimuthal symmetry. In
particular, the radially localized azimuthal modes of perturbed and
irrational-angle aperiodic spirals are extremely attractive for the
engineeringofnovellightsources,laserdevices,andopticalsensors
that combine a broad spectrum of localized eigenmodes with open
dielectric pillar structures for increased refractive index sensitivity
to environmental perturbations.
α
and
β
