Bound Whispering Gallery Modes in Circular Arrays of Dielectric Spherical Particles

Abstract

Low-dimensional ordered arrays of optical elements can possess bound modes having an extremely high quality factor. Typically, these arrays consist of metal elements which have significantly high light absorption thus restricting performance. In this paper we address the following question: can bound modes be formed in dielectric systems where the absorption of light is negligible? Our investigation of circular arrays of spherical particles shows that (1) high quality modes in an array of 10 or more particles can be attained at least for a refractive index nr > 2, so optical materials like TiO2 or GaAs can be used; (2) the most bound modes have nearly transverse polarization perpendicular to the circular plane; (3) in a particularly interesting case of TiO2 particles (rutile phase, nr = 2.7), the quality factor of the most bound mode increases almost by an order of magnitude with the addition of 10 extra particles, while for particles made of GaAs the quality factor increases by almost two orders of magnitude with the addition of ten extra particles. We hope that this preliminary study will stimulate experimental investigations of bound modes in lowdimensional arrays of dielectric particles. 1,2,3,4 They can even serve as a 1 − d nano-waveguide. 5The strong dependence of light scattering and absorption on 1 − d array geometry suggests that the quasi- bound optical modes exist in extended 1 − d arrays. The resonant interaction of light with those modes is responsible for the optical properties of arrays. Each mode can be characterized by the complex eigenfrequency z = ω − iγ/2, with the real part ω representing the mode frequency and the imaginary part γ representing the mode decay rate.6 Modes of interest must have a large enough quality factor Q = ω/γ so that they are bound to the whole 1-d structure.Indeed, 1−d arrays of N identical particles can possess bound modes with a quality factor approaching infinity for N → ∞, 6,7,8,9,10 so γ(N = ∞) = 0. The case Q → ∞ takes place in a linear chain of identical particles separated by a small distance a that is less than half of the resonant wavelength λ a < λ/2 = c/(2ω).(If Eq. (1) is satisfied, then there exists at least one quasibound mode having a maximum wavevector q max = π/a exceeding the wavevector of the resonant photon k = 2π/λ. When the chain is infinite the decay of such mode is forbidden by the momentum conservation law requiring k ≥ q. In the finite system of N particles, the decay rate is finite, but it tends to zero when N → ∞ in accordance with the power law γ(N ) ∝ N −3 (see Ref. 6 ). Quasi-bound modes of the finite system possess a very narrow frequency resonance that can be used in a variety of photonics applications including waveguides, 5 antennas and detectors. 7,8,9 These applications involve the emission or absorption of light in a very narrow frequency range, and the capability of functioning in the single photon regime.11 This narrow resonance leads to a well resolved far field emission pattern that can be used as a guiding signal for aircr...

Optical modes in linear arrays of dielectric spherical particles: a numerical investigation

“…Following from our estimates for circular arrays, 12 the dipolar approach is quite accurate in all cases of interest for the linear chain as well since estimates for multipole interactions up to n max =3 have been considered. We found that if the refractive index exceeds 2, then the corrections to the resonance frequency are less than 2%.…”

“…4 can be therefore understood as the consequence of its small difference from the threshold value. Since corrections to the transversal mode resonant wavevector k were estimated to be around 2% 12 it is not clear whether our preliminary conclusion about longitudinal modes in ZnO would remain valid after more accurate study.…”

“…This contrasts with the circular array behavior, where the quality factor increases with the number of particles exponentially. 9,7,17,12 On our opinion the losses are stronger in finite chains because they have sharp ends where the emission of light takes place. The quality factor dependence on the number of particles is close to the law…”

“…In our other work 12 for instance, we have used the multisphere Mie scattering formalism (3) is written in the frequency domain so all its coefficients depend on the frequency z of the outgoing wave.…”

“…[17], we also ignore the off-diagonal interaction B between the Recall that Mie scattering coefficients are defined by Ricatti-Bessel functions…”