Excitation of nonradiating magnetic anapole states with azimuthally polarized vector beams

Nonradiating current configurations have been drawing the attention of the physics community for many years. It has been demonstrated recently that dielectric nanoparticles provide a unique platform to host such nonradiating modes, called “anapoles”. Here we study theoretically the excitation of such exotic anapole modes in silicon nanoparticles using structured light. Alternative illumination configurations, properly designed, are able to unlock hidden behavior of scatterers. Particularly, azimuthally polarized focused beams enable us to excite ideal anapole modes of magnetic type in dielectric nanoparticles. Firstly, we perform the decomposition of this type of excitation into its multipolar content and then we employ the T-matrix method to calculate the far-field scattering properties of nanoparticles illuminated by such beams. We propose several configuration schemes where magnetic anapole modes of simple or hybrid nature can be detected in silicon nanospheres, nanodisks and nanopillars.


Multipolar decomposition of an arbitrary plane wave in vector spherical harmonics
The electric field produced by an electric dipole p located at r 0 is given by the formula below: Avoiding the singularity at r = r 0 the dyadic Green function above can be decomposed into a series of dyadic products of vector spherical harmonics in the following way [1]: α,−µν (k 0 r 0 ), for r < r 0 α,µν (k 0 r)F (1) α,−µν (k 0 r 0 ), for r > r 0 (S2) We consider a dipole at infinity: k 0 r 0 → ∞ and by applying the large argument approximation of the spherical Hankel functions: z ) radial term as negligible, we end up with the following far field approximation of the dyadic Green function that describes the field that a dipole at infinity, and in a direction given by the unit vectorr 0 (θ 0 , φ 0 ) = r 0 r 0 , produces around the center of the coordinate system: On the other hand, the far-field approximation of the dyadic Green function can also be written as [2]: where r = r − r 0 . If we also restrict again our observation around the center of the coordinate system, we can apply the approximation r r 0 +r·r for the exponential, whereas the approximation r r 0 is sufficient for the denominator. Moreover,r −r 0 . So, the previous expression, under the approximations above, evolves into the following formula: ↔ G f f (r, r 0 ) = e jk 0 r 0 4πr 0 θ 0θ0 +φ 0φ0 e −jk 0r0 ·r .
This last formula actually constitutes the expansion of the cartesian eigensolutions of the Helmholtz equation of free space, that is the plane waves, over the basis of the spherical eigensolutions of it, that is the vector spherical harmonics.

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EBCM method for calculation of the elements of the T-matrix of a rotationally symmetric, star-shaped, homogeneous particle We consider a particle with optical properties ε 1 , µ 1 , k 1 inside an infinite medium with optical properties ε 0 , µ 0 , k 0 . The surface of the particle S 1 is given as a function of the polar and the azimuthal angles r 1 (θ 1 , φ 1 ) in the natural frame of the scatterer. We also define the quantities x 0 = k 0 r 1 , x 1 = k 1 r 1 .
According to the EBCM method [1] the spherical amplitudes of the incident, A α,µν , and the scattered, B α,µν , field are connected with the spherical amplitudes, C α,µν , of the field induced inside the α,µν (k 1 r 1 ), via the Q 3 and Q 1 matrices respectively: (S11) or in a matrix formulation: A = Q 3 · C, B = Q 1 · C. Therefore, the T-matrix will be given by the following formula The elements of the Q q matrices, with index q taking the values of 1 or 3, can be calculated by the following integrals over the surface of the particle:

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where α = β , α = β , n is the perpendicular to the surface of the particle unitary vector and ds n(r 1 ) = r 2 1 sinθ 1r1 − r 1 sinθ 1 ∂ r 1 ∂ θ 1θ 1 − r 1 ∂ r 1 ∂ φ 1φ 1 dθ 1 dφ 1 . For a rotationally symmetric particle ∂ r 1 ∂ φ 1 = 0 we can perform the integration over the azimuthal angle analytically and therefore we end up with the simplified formulas Q M,µν;(q) M,µ ν Finally, for the particular case of a rotationally symmetric cylindrical disk of height h, diameter d and aspect ratio A r = d h , the surface of the particle will be given by the function with its partial derivative over the polar angle being

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Additional plots supporting the results of Figure 1 of the main manuscript Figure S1: a) Phase mask proposed to eliminate the magnetic octupole interference that hinders the access to the anapole condition. b,c) Multipolar decomposition of the normalized scattered power P sca /P inc corresponding to two illumination schemes: single beam excitation with phase mask applied and excitation with two out-of-phase beam, respectively, the combination of which gives us the excitation of the ideal magnetic anapole of Figure 1c of the paper.
Field plots that correspond to Figure 1b,c of the main manuscript Figure S2: Normalized Electric and Magnetic field intensity plots that refer to single-and two-beam excitation schemes, with and without the proposed phase modulation, of a silicon sphere of size parameter x 0 = 1.62 placed at the focal point and corresponding to the anapole condition case. The field plots are over a region of 2 × 4 wavelengths at the focal area on the ρOz plane. Under the single beam excitation scheme we have both quadrupole and octupole terms spoiling the anapole condition.
Adding a second, out of phase, counterpropagating beam we manage to kill the quadrupole interference, and, moreover, applying the proposed phase mask we elliminate also the octupole interference, leading finally to an ideal magnetic anapole excitation. The major price that we pay is that the phase mask weakens the field intensity at the vicinity of the focal point.