Optical theorem and multipole scattering of light by arbitrarily shaped nanoparticles

The application of Cartesian multipoles in irreducible representations provides the possibility to get explicit contributions of the toroidal multipole terms in the extinction and scattering power without the introduction of special form factors. In the framework of the Cartesian multipoles, we obtained multipole decomposition (up to the third order) of the induced polarization (current) inside an arbitrarily shaped scatterer (nanoparticle). The third-order decomposition includes the toroidal dipole, magnetic quadrupole, electric octupole terms, and also nonradiating terms. The corresponding multipole decomposition of the scattering cross section, taking into account the electric octupole term, is derived and compared with the multipole decomposition of the extinction cross section obtained using the optical theorem. We show that the role of multipoles in the optical theorem (light extinction) and scattering by arbitrarily shaped nanoparticles can be different. This can result in seemingly paradoxical conclusions with respect to the appearance of multipole contributions in the scattering and extinction cross sections. This fact is especially important for absorptionless nanoparticles, for which the scattering cross section can be calculated using the optical theorem, because in this case extinction is solely determined by scattering. Demonstrative results concerning the role of third-order multipoles in the resonant optical response of high-refractive-index dielectric nanodisks, with and without a through hole at the center, are presented. It is shown that the optical theorem results in a negligible role of the third-order multipoles in the extinction cross sections, whereas these multipoles provide the main contribution in the scattering cross sections.

Figure 1

(a), (b) Extinction cross-section spectra for a disk shaped dielectric nanoparticle, with a radius of 150 nm and height of 60 nm, located in air. (a) The multipole decomposition corresponds to expression (39). (b) The multipole decomposition corresponds to expression (40). (c) Scattering cross-section spectra for the same nanoparticle. The multipole decomposition corresponds to expression (36). Incident light is a linearly polarized plane wave propagating perpendicular to the disk surface. The graphs also show contributions from different multipoles.

Figure 2

Scattering directivity (arbitrary units) for the same disk nanoparticle and irradiation conditions (see Fig. 1) at 674 nm wavelength in the spherical coordinates. The diagram center $\left(\theta =0\right)$ corresponds to forward scattering; the diagram edges $\left(\theta =\pi \right)$ correspond to backward scattering. (a) Total scattering and (b) MQ scattering diagrams.

Figure 3

(a) Extinction cross-section spectra for the disk shaped silicon [22] nanoparticle, with a radius of 150 nm and height of 60 nm having a through hole at the center with a radius of 75 nm, located in air. (a) The multipole decomposition corresponds to expression (39). (b) The multipole decomposition corresponds to expression (40). (c) Scattering cross-section spectra for the same nanoparticle. Irradiation conditions are the same as in Fig. 1. The graphs also show contributions from different multipoles.