## Abstract

The existence of non-radiating sources is of fundamental importance for inverse scattering problems and the design of invisible objects. However, the excitation of such radiationless sources is quite challenging. We present a method based on which the anapole mode of a high-index isotropic dielectric nanosphere can be excited but radiationless. We show that this radiationless anapole is attributed to the destructive interference of the Cartesian dipole and toroidal moment of the induced current by our proposed focused radially polarized beam illumination. Further, with a standing-wave illumination formed by two counter-propagating focused radially polarized beams under $4\pi $ configuration, the ideal radiationless anapole can be excited. This result illustrates a case where the reciprocity condition is not violated, and yet, a radiationless mode can be excited by external illumination.

© 2016 Optical Society of America

Objects that cannot be seen have been a fascinating possibility both in popular imagination and in the scientific community. Beside the practical implications in modern-day security applications, it is also of great interest in the general scientific study of the universe [1]. There are different possible types of these objects, which are radiationless. These can be objects that do not scatter incident light [2]. There also can be time-varying charges or current distributions or atoms that do not radiate [3–6]. The study of such non-radiative objects has been a part of fundamental physics for a long time [7–9]. Studies have aimed to understand the physics of these objects from scattering from non-absorbing particles [2], inverse scattering problems [5,10], and metamaterials to suppress radiation loss [11,12], to mention a few. Possible applications of this physics for designing non-scattering objects are also being investigated [13].

One such case of non-radiating sources is the anapole. This was first introduced in elementary particle physics [14] and recently has been of interest in particle physics again [1]. The anapole mode arises when the electrical dipole and toroidal dipole moment form a non-trivial destructive interference [15–17]. However, it was not until recently that the non-radiating anapole mode was experimentally demonstrated [18,19]. In the Mie scattering theory, under certain conditions, a specific multipole mode from the multipole decomposition of the scattering field of an illuminated particle can be completely suppressed. For example, the scattering coefficient of the spherical electric dipole moment can be zero at certain wavelengths for a high-index dielectric nanosphere under plane wave illumination [20]. However, at the same time, a strong and broadband magnetic dipole resonance can be excited in the high-index dielectric nanosphere [21,22]. Therefore, the non-radiating anapole mode of a nanosphere is difficult to observe under plane wave illumination due to the simultaneous excitation of the electric and magnetic modes. That is why the first experimental demonstration of an anapole in the visible wavelength range [19] was done with a specially designed nanodisk instead of a sphere. In spite of constructing the nanodisk for the specific anapole condition, the scattering from the particle cannot be completely reduced to zero due to the simultaneous excitation of a magnetic quadrupole mode. Instead of specifically designed structures [19,23], in this work, we propose an alternative way to excite the anapole mode of an isotropic nanosphere with tightly focused radially polarized beams. The focused radially polarized beam, which has a pure longitudinally polarized electric field ${E}_{z}$ and zero magnetic field at the focal point [24], can keep all the magnetic modes suppressed and efficiently induce a current that does not radiate electric dipole mode at the anapole condition except for a weak electric quadruple. In addition, we show that when two focused radially polarized beams under the $4\pi $ configuration illuminate the nanosphere, the electric quadruple radiation is completely suppressed, and an ideal radiationless anapole is excited.

The configuration of the proposed anapole excitation is shown in Fig. 1(a), where a tightly focused radially polarized beam illuminates a high-index dielectric nanosphere of radius ${r}_{0}=100\text{\hspace{0.17em}}\mathrm{nm}$ and refractive index $n=3.5$, placed at the focal point. The surrounding medium is air, but the result holds for any high-index sphere embedded in a low-index, homogeneous, and isotropic medium. In the numerical implementation, the focal field of the radially polarized beam is calculated by the Richard–Wolf diffraction integral [25] and then imported to a finite element method (FEM) simulation [26] to calculate the scattering properties of the nanosphere. In Fig. 1(b), we show the calculated spectral dependence of the scattering power and the internal energy of the particle. Here, the electrical energy inside the particle ${W}_{\mathrm{E}}=\frac{1}{2}{\int}_{\text{sph}}({\mathbf{EE}}^{*})\mathrm{d}\mathbf{r}$ is normalized to the focal energy ${W}_{\mathrm{f}}=\frac{1}{2}{\int}_{\text{sph}}({\mathbf{E}}_{\mathbf{f}}{\mathbf{E}}_{\mathbf{f}}^{*})\mathrm{d}\mathbf{r}$ within the same volume of the sphere $\text{sph}=\{\mathbf{r}\text{:}|\mathbf{r}|\le {r}_{0}\}$ when the particle is absent. We can clearly see from the scattering power spectrum in Fig. 1(b) that only electric modes are excited, while the magnetic modes, including the magnetic quadruple resonance at $\lambda =505\text{\hspace{0.17em}}\mathrm{nm}$, are suppressed. This selective excitation of electric modes by focused radially polarized beam has been shown in previous works [27–29] and can be explained explicitly by the multipole expansion of the focal field [30]:

In order to decide that whether an anapole moment is excited, the field inside the particle is numerically calculated and Cartesian multipole moment analysis [19] is applied on the induced current,

inside the particle. The focused radially polarized beam induces an inhomogenous current distribution inside the particle. It excites not only the Cartesian dipole moment, but also the toroidal dipole momentThe toroidal dipole moment, featuring an electric poloidal current with a circulating magnetic field, has exactly the same far-field scattering pattern as a Cartesian dipole moment. An anapole is excited if the condition $\mathbf{P}=-ik\mathbf{T}$ is fulfilled, where the electrical dipole moment and toroidal moment interfere destructively with each other, resulting in the complete cancellation of their far-field radiation. The $z$ component of the Cartesian dipole moment $\mathbf{P}$ and toroidal moment $ik\mathbf{T}$ of the high-index nanosphere in Fig. 1(a) are calculated numerically and illustrated in Fig. 2. Due to cylindrical symmetry around the optical axis of both the focal field of radially polarized beam and the nanosphere, the induced current $\mathbf{J}$ is also cylindrically symmetric, which can be seen from the inserted figure about the electric field distribution inside the particle in Fig. 2(a). As a result of this symmetry, both the electric dipole moment $\mathbf{P}$ and the toroidal moment $\mathbf{T}$ have only the $z$ component as a non-zero component. The minimums in the scattering power seen in Fig. 1 at $\lambda =464\text{\hspace{0.17em}}\mathrm{nm}$ are explained by the wavelength-dependent plots of the electric dipole moment $\mathbf{P}$ and the toroidal moment $\mathbf{T}$ seen in Fig. 2. Here, we observe that at this wavelength, the anapole mode condition ${P}_{z}=-ik{T}_{z}$ is fulfilled such that the scattering fields of dipole and toroidal moment have the same strength but are out of phase, resulting in the complete cancellation of the far-field dipole mode radiation.

From Fig. 1(b), we can see that even at the anapole condition $\lambda =464\text{\hspace{0.17em}}\mathrm{nm}$, the scattering is not completely suppressed. This is confirmed by the radiation pattern in Fig. 3. Here we clearly see that though the peak scattering power at $\lambda =464\text{\hspace{0.17em}}\mathrm{nm}$ is 40 times weaker compared to the electric dipole resonance at $\lambda =554\text{\hspace{0.17em}}\mathrm{nm}$, it is not zero. Since both the electric dipole and toroidal moments have only the $z$ component, the strength of the far-field dipole radiation can be evaluated in the $z=0$ plane. As shown in Fig. 3(a), the scattering power in the $z=0$ plane is zero at $\lambda =464\text{\hspace{0.17em}}\mathrm{nm}$. Therefore, though the far-field dipole radiation of the induced current in the nanosphere is canceled by the destructive inteference of the Cartesian dipole moment and toroidal moment, a weak electric quadruple mode is excited at the same wavelength and results in non-zero total scattering.

In a second example, we show that an ideal radiationless anapole can be realized with an excitation field formed by the interference of two counter-propagating focused radially polarized beams under a $4\pi $ configuration, as illustrated in Fig. 4. The two beams have exactly the same amplitude distribution but a $\pi $ phase difference. Based on Eq. (10) in Ref. [30], we can get the strength of electric multipoles for the $4\pi $ excitation field:

In recent literature, it has been speculated whether it is possible to achieve total zero scattering using external sources [19] due to the possible violation of the reciprocity theorem. So to examine our proposal for the ideal radiationless anapole under $4\pi $ illumination, we consider the following. Let the focal field of the $4\pi $ configuration in the absence of the particle, ${\mathbf{E}}_{\mathbf{f}}^{4\pi}$, be generated by a current ${\mathbf{J}}_{\mathbf{f}}^{4\pi}$ in the source volume $V$. Additionally, let ${\mathbf{E}}_{\mathbf{s}}$ be the electric field radiated by the induced current $\mathbf{J}=-i\omega ({n}^{2}-1)\mathbf{E}$ in the nanosphere. According to the reciprocity theorem [32], we should have

For the ideal non-radiating current source $\mathbf{J}$, the scattering far field ${\mathbf{E}}_{\mathbf{s}}$ in source volume $V$ should be zero, so that implies ${\int}_{V}{\mathbf{J}}_{\mathbf{f}}^{4\pi}\xb7{\mathbf{E}}_{\mathbf{s}}\mathrm{d}\mathrm{r}=0$. Complying with the reciprocity theorem only requires that the integral ${\int}_{\text{sph}}{\mathbf{E}}_{\mathbf{f}}^{4\pi}\xb7\mathbf{J}\mathrm{d}\mathrm{r}$ be zero instead of having zero induced current everywhere inside the source volume. It has been numerically verified that this condition is fulfilled by the radiationless anapole mode of the nanosphere excited by the $4\pi $ illumination, with the extremely weak electric octupole component neglected.

In summary, we demonstrate that it is possible to excite the anapole mode of an isotropic high-index dielectric nanosphere by illumination with tightly focused radially polarized beams. With single focused beam excitation, the anapole dipole mode can be excited, but there is still a weak contribution of electric quadruple radiation to the total scattering power. However, an ideal radiationless anapole can be excited by a $4\pi $ configuration, where the two counter-propagating radially polarized beams have the same amplitude but a $\pi $ phase difference, illustrating a condition where the reciprocity condition is not violated, and yet, a non-radiating mode can be excited by external illumination. This is the first time to our knowledge that a configuration has been found where the non-radiating anapole mode can be excited inside an un-engineered isotropic spherical Mie particle. The anapole moment has to be studied further to understand the physics of the nonscattering objects, non-radiating sources, and as an electromagnetic form factor of particles. Applications of this mode in non-invasive sensing, suppression of spurious scattering, design of invisible objects, and optical switches where the scattering can be switched on and off with the anapole mode could be possible in the future.

## Funding

NanoNextNL.

## Acknowledgment

This work is supported by NanoNextNL of the Government of the Netherlands and 130 partners.

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