Cylindrical vector beams reveal radiationless anapole condition in a resonant state

Nonscattering optical anapole condition is corresponding to the excitation of radiationless field distributions in open resonators, which offers new degrees of freedom for tailoring light-matter interaction. Conventional mechanisms for achieving such a condition relies on sophisticated manipulation of electromagnetic multipolar moments of all orders to guarantee superpositions of vanished moment strengths at the same wavelength. In contrast, here we report on the excitation of optical radiationless anapole hidden in a resonant state of a Si nanoparticle utilizing tightly focused radially polarized (RP) beam. The coexistence of magnetic resonant state and anapole condition at the same wavelength further enables the triggering of resonant state by tightly focused azimuthally polarized (AP) beam whose corresponding electric multipole coefficient could be zero. As a result, high contrast inter-transition between radiationless anapole condition and ideal magnetic resonant scattering can be achieved experimentally in visible spectrum. The proposed mechanism is general which can be realized in different types of nanostructures. Our results showcase that the unique combination of structured light and structured Mie resonances could provide new degrees of freedom for tailoring light-matter interaction, which might shed new light on functional meta-optics.

transition between radiationless anapole condition and ideal magnetic resonant scattering can be achieved experimentally in visible spectrum. The proposed mechanism is general which can be realized in different types of nanostructures. Our results showcase that the unique combination of structured light and structured Mie resonances could provide new degrees of freedom for tailoring light-matter interaction, which might shed new light on functional meta-optics.
According to Mie theory, the total scattering power of a spherical nanoparticle excited by a plane wave is determined by both contributions of electric and magnetic multipole moments of different order n 1 where E i , k and w are the amplitude, k-vector and angular frequency of the incident plane wave, respectively.There are numerous zeros of Mie scattering coefficients a n and b n , which are corresponding to zero contribution to the total scattering from the electromagnetic multipole moments of different orders 25 . It was suggested that the electromagnetic anapole condition could be interpreted as the destructive interference between toroidal moments and Cartesian electromagnetic multipole moments 4-7, 14, 20, 26 . In general, the zero amplitude conditions of a n and b n are usually accompanied with other spectral overlapping electromagnetic multipole moments. As a result, such zero conditions of Mie scattering coefficients could be physically hidden in far-field scattering response. For example, zero |a 1 | can even coexist with the resonant magnetic dipole (MD) condition at the same frequency 25 .

Mechanism and numerical results
The electric field of a focused CVB can be expressed in terms of the electric and magnetic multi- where p 0 El and p 0 M l are the strength of the electric and magnetic multipole components, N 0 l and M 0 l are the vector spherical harmonics related to the electric and magnetic multipole components, re-spectively. For a RP beam without orbital angular momentum, the magnetic component p 0 M l is zero while the electric component p 0 El is zero for a AP beam 44 . As a result, the focused RP beam can be used to excite the electric and toroidal dipole moments because of their similarity of far-field radiation 9, 16, 37 while the focused AP beam can be used to excite the magnetic multipole modes [31][32][33][34][35][36] . As the excited strength of an electromagnetic multipolar mode depends on both the vectorial properties of the excitation source and the eigenmodes supported by the spherical nanoparticle, therefore, the key enablers for realizing high contrast reconfigurable optical scattering include two points:1) a nanoparticle supports a pure electromagnetic multipole mode at a specified frequency, which is spectrally overlap with an anapole condition; 2) the spatial overlapping between the electromagnetic field of the nanoparticle's eigenmode and that of the tightly focused CVBs. High permittivity dielectric nanoparticles resemble a promising platform that fulfills both conditions. For a spherical In order to show the generality of this mechanism, we further consider a simple dielectric nanostructure where its magnetic quadrupole resonance overlaps with the anapole condition. Such dielectric nanostructure is much easier to fabricate in experiment than the core-shell nanoparticle.
The Si nanodisk is optimized to fulfill the aforementioned enablers to realize high contrast recon-

Experimental results
The Si nanodisk shown in Figure 3 can be readily fabricated on a glass substrate (see Supplement 1). The substrate effect only has minor effect on the results of Figure 3, as will be addressed in the following. The geometry parameters of the Si nanodisk are outlined in Figure 5a, where the radius r of the Si nanodisk is 150 nm while the height h is 160 nm. The top and side views of scanning electron microscopy (SEM) images of the fabricated Si nanodisk are shown in Figure   5b and c, respectively. The side view is taken by tilting the sample stage of SEM by 30 degree.
As can be seen in Figure 5e, the back-scattering spectra measured by the home-built optical setup shown in Figure 5d features a resonant scattering at around 735 nm under the excitation of tightly focused AP beam. The resonance wavelength is red-shifted compared with the result of Figure 3d because of the substrate effect (see Supplement 1). However, the scattering intensity at the same wavelength is one order of magnitude smaller than the AP case when a tightly focused RP beam is used. This results quantitatively agree with the theoretical results of

Conclusion
In summary, we propose a new mechanism to excite the radiationless anapole condition without