Journal of Nanomaterials

Volume 2015, Article ID 672957, 7 pages

http://dx.doi.org/10.1155/2015/672957

## Elusive Pure Anapole Excitation in Homogenous Spherical Nanoparticles with Radial Anisotropy

^{1}College of Optoelectronic Science and Engineering, National University of Defense Technology, Changsha, Hunan 410073, China^{2}Nonlinear Physics Centre, Research School of Physics and Engineering, Australian National University, Acton, ACT 0200, Australia

Received 30 August 2015; Accepted 18 October 2015

Academic Editor: Christian Brosseau

Copyright © 2015 Wei Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For homogenous isotropic dielectric nanospheres with incident plane waves, Cartesian electric and toroidal dipoles can be tunned to cancel each other in terms of far-field scattering, leading to the effective anopole excitation. At the same time however, other multipoles such as magnetic dipoles with comparable scattered power are simultanesouly excited, mixing with the anopole and leading to a nonnegligible total scattering cross-section. Here, we show that, for homogenous dielectric nanospheres, radial anisotropy can be employed to significantly suppress the other multipole excitation, which at the same time does not compromise the property of complete scattering cancallation between Cartesian electric and toroidal dipoles. This enables an elusive* pure anopole* excitation within radially anisotropic dielectric nanospheres, which may shed new light on many scattering related fundamental researches and applications.

#### 1. Introduction

Since the discovery by J. J. Thomson that electrons are constituents of atoms [1], there had been a lot of searches for charge-current distributions in which no electromagnetic energy is radiated, which can be applied to explain the stability of atoms [2]. This is due to the existing dogma since the establishment of Maxwell equations that accelerating charges would lose energy continuously through electromagnetic radiation, and consequently to prevent the collapse of atoms, the charges within them must be arranged into specific configurations. Such special charge-current configurations are also termed as nonradiating sources [2, 3]. Those nonradiating sources are deeply related to the concepts of anapoles in quantum field theory, which have their roots in the law of parity unconservation [4], and can be used as possible explanations for the existence and features of neutrinos and cold dark matter [5, 6].

For time-varying oscillating charge-current distributions, the dynamic version of anapoles is highly related to the so-called toroidal multipoles, which recently have attracted enormous attention [7–20]. As independent terms in the multipole expansion of electromagnetic fields besides conventional cartesian magnetic and electric multipoles [3], toroidal multipoles correspond to significantly different near-field charge-current distributions and will inevitably become significant when the operating spectral regime goes beyond the electrostatic region [20–22]. In the far field, however, the toroidal multipoles have identical scattering patterns compared with their conventional Cartesian electric and magnetic counterparts, and that is to say, they are indistinguishable in terms of far-field scattering [7, 13, 21, 23]. This at the same time also means that when properly engineered, the contributions of conventional and toroidal multipoles to the scattered field can be tuned to the same amplitude and out of phase, which leads to the far-field scattering cancellation, and thus generate the so-called electrodynamic anapoles [11, 15–17, 19, 21, 24]. The simplest case of such anapoles is induced by the destructive cancellation of Cartesian electric dipole and toroidal dipole, which has been demonstrated firstly in composite carefully engineered structures consisting of particle clusters [11, 15]. Recently it has also been theoretically proposed and experimentally demonstrated in surprise that the anapole can be effectively excited even in individual homogenous dielectric nanoparticles [16]. However, the problem of the above demonstration in simple structures is that, besides the anapole excitation, other multipoles (such as magnetic dipoles) of similar magnitudes in terms of scattered power are also excited and mixed with the excited electric and toroidal dipoles [16, 17]. Such problems can be conquered within single core-shell nanoparticles of fundamental shapes (spherical or cylindrical) where the extra multipole excitations can be significantly suppressed, enabling elusive* pure anapole* excitations [17, 19].

Here we propose an alternative approach and demonstrate that elusive* pure anapole* can be observed in homogenous radially anisotropic dielectric spherical nanoparticles under incident plane waves. It is shown that when the radial anisotropy of material permittivity is employed, the Cartesian electric and toroidal dipoles can be tuned to the spectral positions where other multipoles are significantly suppressed, without compromising the feature of complete destructive interference between them. Since Cartesian electric and toroidal dipoles can totally cancel each other in terms of far-field scattering and at the same time the excitation of other multipoles is negligible, it is proved that effectively* pure anapole* has been obtained. The anapole excitation is highly related to the exotic scattering phenomena of invisibility and possible applications of noninvasive sensing and detections, which may pave the way for many further scattering related fundamental researches and applications.

#### 2. Methods and Expressions

##### 2.1. Multipole Expansions with Toroidal Components

For any charge-current distribution, when the geometric size of the source is far smaller than the effective wavelength (the retardation effect within the source can be neglected), the quasi-static approximation can be applied and then the radiated fields of the source can be expanded into two sets of conventional Cartesian electric and magnetic multipoles [3, 21]. The lowest order terms of the two sets of multipoles are electric and magnetic dipoles, which can be expressed, respectively, as (throughout the paper we adopt the notation for electromagnetic waves)where is the speed of light, is the angular frequency, and is the current. When the retardation effect within the source cannot be neglected (the size of the source gets larger and/or the effective wavelength gets smaller), the two sets of multipoles are not complete and thus other multipoles have to be included as correcting terms [20–22]. The lowest order correcting multipoles are toroidal multipoles, with the toroidal dipole expressed as [7, 21, 23, 24]

The radiated power of , **,** and is, respectively, where is the vacuum permeability and is the angular wave number. We emphasize that the expressions above are for individual multipoles only and a simple sum of them does not lead to the total scattered power as there is interference between them [7, 21, 23] (see also (8) below). It is also worth mentioning that the incorporation of toroidal multipoles does not make the expansion complete [20–22]. When the operating spectral region is far beyond the quasi-static regime, besides toroidal multipoles, other higher order correcting multipoles have to be also included [20–22].

For scattering particles, based on the charge conservation relation, the current (displacement current) distribution iswhere is the vacuum permittivity; is the relative permittivity, which is a scalar for isotropic particles or tensor for anisotropic particles. With the current distribution shown in (4), all the Cartesian multipoles including , **,** and and their scattered power can be directly calculated (see (1)–(3)).

##### 2.2. Electric and Magnetic Multipoles Deduced from Vector Spherical Harmonics of Mie Scattering Particles

For spherical (both isotropic and radially anisotropic) particles with incident plane waves of electric field and magnetic field , both the fields inside the particle and the scattered fields can be expressed in transverse electric and transverse magnetic (in terms of electric field distribution) vector spherical harmonics [20, 25–28]. On one hand, according to (1)–(4), all the Cartesian multipoles including , , and and their scattered power can be analytically expressed and numerically calculated. On the other hand, based on the far-field equivalence, the scattered fields can be viewed as the radiated fields of a combination of conventional Cartesian electric and magnetic multipoles (which correspond to transverse magnetic and transverse electric vector spherical harmonics, resp.). As a result the far-field deduced electric dipole and magnetic dipole can be expressed, respectively, as [16, 17, 20, 25, 28–32]where and are the Mie scattering coefficients of the lowest order. The total scattered power is [25–28]where and are the Mie scattering coefficients of order and . It is clear that the scattered power of and is, respectively,

##### 2.3. Anapole Excitation Induced by Destructive Interference Between and

As is discussed above, shown in (5) has been obtained based on the complete vector spherical harmonic expansion and thus has included all the transverse magnetic dipolar scattering components [16, 17, 20, 28–32], while the conventional Cartesian electric dipole shown in (1) is calculated through current integration and is only the lowest order expansion of [20–22]. For operating spectral region beyond the quasi-static regime, the next order correcting term, which is the toroidal dipole shown in (5), should also be included. When higher order correcting terms beyond toroidal terms can be neglected, the scattered dipolar transverse magnetic field can be expressed as [16, 17, 21, 23] where is the unit vector along . The existence of the anapole lies in which can be satisfied when

It is worth mentioning that when (9) is satisfied, only the transverse magnetic dipolar scattering will be vanishing and consequently the anapole excited (consisting of destructively interfering and ) can be mixed with other excited multipoles, such as magnetic dipoles, electric quadrupoles, and so on [16, 17, 19].

#### 3. Results and Discussions

In Figure 1 we provide a schematic illustration of anopole excitation in a homogenous particle with plane wave incidence. The plane wave is polarized along direction (in terms of electric field) and propagating along direction. The radius of the particle is , and the particle is isotropic with refractive index or is radially anisotropic with transverse (perpendicular to the radial direction) permittivity and radial permittivity , with the anisotropy parameter defined as . Both and are excited, out of phase and of the same magnitude with regard to the scattered power, indicating excitation of the anapole ((8)-(9)). Also the typical current and magnetic field distributions for only have been shown.