Maximizing the Electromagnetic Chirality of Thin Dielectric Tubes

Abstract

Any time-harmonic electromagnetic wave can be uniquely decomposed into a left and a right circularly polarized component. The concept of electromagnetic chirality (em-chirality) describes differences in the interaction of these two components with a scattering object or medium. Such differences can be quantified by means of em-chirality measures. These measures attain their minimal value zero for em-achiral objects or media that interact essentially in the same way with left and right circularly polarized waves. Scattering objects or media with positive em-chirality measure interact qualitatively different with left and right circularly polarized waves, and maximally em-chiral scattering objects or media would not interact with fields of either positive or negative helicity at all. This paper examines a shape optimization problem, where the goal is to determine thin tubular structures consisting of dielectric isotropic materials that exhibit large measures of em-chirality at a given frequency. We develop a gradient based optimization scheme that uses an asymptotic representation formula for scattered waves due to thin tubular scattering objects. Numerical examples suggest that thin helical structures are at least locally optimal among this class of scattering objects.

Maximizing the electromagnetic chirality of thin metallic nanowires at optical frequencies

“…In addition to the shape of the spine curve of the nanowire, we also optimize a possible twisting of its cross-section along the spine curve. This extends an earlier study in [6], where a free-form shape optimization for thin dielectric nanowires with circular crosssections has been discussed. It has been observed in [6] that the optimized thin dielectric nanowires do not possess very high values of em-chirality at optical frequencies.…”

“…This extends an earlier study in [6], where a free-form shape optimization for thin dielectric nanowires with circular crosssections has been discussed. It has been observed in [6] that the optimized thin dielectric nanowires do not possess very high values of em-chirality at optical frequencies. The distinguishing feature of the noble metals considered for the nanowires in this work, in particular of silver and gold, is the negative real part of their electric permittivity at optical frequencies combined with a relatively small positive imaginary part.…”

“…Proof. Using Lemma 4.2, this result can be shown as in [6,Thm. 4.2], where the Fréchet differentiability of T ρ for thin dielectric nanowires with circular cross-sections has been established.…”

“…In Lemma 3.2 and Remark 4.3 of [6], formulas for the coefficients in the series representations of T ρ (p, V p,θ ) and of its Fréchet derivative T ′ ρ [p, V p,θ ](h, φ) in terms of these circularly polarized vector spherical harmonics have been developed for the corresponding operators associated to thin dielectric nanowires with circular cross-sections. Observing that the different material properties and the more general twisting cross-sections considered in this work only affect the electric polarization tensor and its Fréchet derivative in T ρ (p, V p,θ ) and T ′ ρ [p, V p,θ ](h, φ), the results from [6] can immediately be applied to obtain formulas for the coefficients in the series representations of the operators considered in the present work as well.…”

“…Therewith, we develop a quasi-Newton algorithm that does not require to solve a single Maxwell system during the optimization procedure. This shape optimization scheme is an extension of an algorithm from [6]. The novel features are that we consider metallic nanowires, that we allow for arbitrary cross-sections, and that we optimize not only the shape of the spine curve of the nanowire but also the twisting of the cross-section of the nanowire along the spine curve.…”