# Cylindrical vector beams reveal radiationless anapole condition in a
resonant state

###### Abstract

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.

Yudong Lu 1, Yi Xu 1,2,*, Xu Ouyang 1,2, Mingcong Xian 2, Yaoyu Cao 1,Kai Chen
1 , and Xiangping Li 1,*

${}^{\textit{1}}$ Guangdong Provincial Key Laboratory of Optical Fiber Sensing
and Communications, Institute of Photonics Technology, Jinan University,
Guangzhou, 510632, China

2Department of Electronic Engineering, College of Information Science and
Technology, Jinan University, Guangzhou, 510632, China

*Corresponding author<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1 Introduction

Optical scattering of a nanopaticle under the excitation of a plane wave is
usually determined by its predominant electromagnetic multipole moment [1].
Such predominant multipole moment can even decide the electric or magnetic
nature of the scattering in all-dielectric photonics [2]. Therefore, tailoring
such predominant electromangic multipole moment becomes an effective and
unified way to manipulate optical scatteirng [3]. Many abnormal optical
scattering phenomena have been proposed to enable new possibility and
functionality in photonics [3]. The nonscattering electromagnetic state is one
of the typical examples [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19] [see these elaborated reviews in Refs. [20, 21, 22, 23] for details]. In
contrast to the embedded eigenstates or bound state in the continuum which
cannot be accessed by the excitation in the continuum, the so-called
electromagnetic anapole condition provides a nonscattering condition sustained
under the excitation of external field[46, 47], which resembles a promising
physical mechanism for tailoring light-matter interaction in a nonscattering
manner [20, 21, 22, 23, 24]. It is generally perceived that such anapole
condition requires that all of the induced electromagnetic multipole moments
vanish at the same wavelength, resembling the canonical way of realizing the
radiationless anapole condition [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
17, 18, 19]. In particular, the optical anapole condition corresponds to a
pronounced minimum in the far-field scattering associated with highly confined
electromagnetic near-field 6, 26. Such condition is quite challenge to meet
since the degree of freedom for tailoring the induced electromagnetic
multipolar moments of a nanoparticle is quite limited. Such a condition can
also be readily fulfilled through structured light illumination [16] which
generally extends the scopes of light-matter interaction from both fundamental
science and application prospectives, such as high numerical aperture (NA)
focusing [27, 28, 29], optical computation[45], optical data storage [30],
customized excitation of electromagnetic multipole resonances [31, 32, 33, 34,
35, 36, 37] and radiationless anapole condition [9, 16, 17], enhancement of
optical nonlinearity [38, 39], optical tweezers [40, 41] and advanced
metrology [42, 43].

Figure 1:  Schematic of a reconfigurable optical antenna which supports the
radiationless anapole condition hidden in a magnetic resonances at the same
frequency. Tightly focused RP (a) or AP (b) beam is used to selectively
realize the nonscattering and resonant scattering scenarios of the optical
antenna.

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]

$\displaystyle
P_{total}=\frac{\pi|E_{i}|^{2}}{k\omega\mu_{0}}\sum_{n=1}^{\infty}(2n+1)(|a_{n}|^{2}+|b_{n}|^{2})$
(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, 5, 6, 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].

In this paper, we show that sophisticated tailoring of electromagnetic
multipolar moments in nanoparticles is not essential for the realization of
the anapole condition, which is different from previous efforts in realizing
the anapole condition[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19]. In particular, we experimentally demonstrate that the combination of
tightly focused cylindrical vector beam (CVB) and structured Mie resonances
can reveal the radiationless anapole condition which is overlapped in spectrum
with a resonant state. More importantly, it subsequently enables
reconfigurable optical scattering of a silicon nanoparticle with simple
morphology, where the optical scattering of the nanoparticle at a specified
wavelength can be switched from the radiationless anapole condition to the
magnetic resonant scattering condition and vice versa, as shown in Fig. 1.

##  2 Mechanism and numerical results

Figure 2:  Numerical results of electromagnetic multipolar decomposition for
the normalized scattering power of a Au core/Si shell nanoparticle under the
excitation by a plane wave (a), a tightly focused AP beam (b) and a tightly
focused RP beam (c). The radius of Au core is 86 nm while the outer radius of
Si shell is 226 nm. The NA and magnification factor of the objective lens are
0.95 and 60, respectively. TED represents the contribution of toroidal and
electric dipole to total scattering.

The electric field of a focused CVB can be expressed in terms of the electric
and magnetic multipole fields [44]

$\displaystyle
E_{f}(\textbf{r})=\sum_{l=1}^{\infty}\sum_{-l}^{l}[p_{El}^{0}\textbf{N}_{l}^{0}(\textbf{r})+p_{Ml}^{0}\textbf{M}_{l}^{0}(\textbf{r})]$
(2)

where $p_{El}^{0}$ and $p_{Ml}^{0}$ are the strength of the electric and
magnetic multipole components, $\textbf{N}_{l}^{0}$ and $\textbf{M}_{l}^{0}$
are the vector spherical harmonics related to the electric and magnetic
multipole components, respectively. For a RP beam without orbital angular
momentum, the magnetic component $p_{Ml}^{0}$ is zero while the electric
component $p_{El}^{0}$ 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 Au core/Si shell nanoparticle (SN), the contributions of the
spherical electric dipole (ED) can be designed to be totally suppressed at the
resonant condition of MD [25]. One of the typical solution under plane wave
excitation is shown in Figure 2a. If the scattering of a tightly focused AP
beam is considered, it can be seen from Figure 2b that all electric multipole
moments have zero contribution to the total scattering, resembling the ideal
MD scattering. On the contrary, the total scattering is zero at the same
frequency of the MD resonance if the excitation of a tightly focused RP beam
is applied, realizing the optical anapole condition, as shown in Fig. 2c. Such
a sharp contrast in optical scattering power of a nanoparticle at the same
wavelength enables the realization of reconfigurable optical scattering.

Figure 3:  Cartesian electromagnetic multipolar decomposition results for the
scattering power of a Si nanodisk under the excitation of a tightly focused RP
beam (a) and a tightly focused AP beam (d), respectively. Multipolar moments
up to quadrupole are considered. The radius of Si nanodisk is r = 150 nm while
its height is h = 160 nm. The corresponding electric and magnetic field
distributions at several cross sections are shown in (b), (c) and (e), (f)
under the anapole condition and magnetic quadrupole (MQ) resonance,
respectively. All the wavelength are 720nm. The NA and magnification factor of
the objective lens are 0.95 and 60, respectively.  Figure 4:  Evolution of the
total scattering spectra for the Si nanodisk with different radius under the
excitation of RP (a) and AP (b) beams with fixed focusing properties. The
dependence of the anapole condition and MQ resonance on the radius of the Si
nanodisk are outlined by dashed lines. The NA and magnification factor of the
objective lens are 0.95 and 60, respectively.

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 reconfigurable optical scattering in the visible spectrum. According
to the Cartesian electromagnetic multipole decomposition results under the
excitation of a focused RP beam shown in Figure 3a, the destructive
interference between the Cartesian ED and toroidal dipole moments results in
the anapole condition where the contributions of high order electromagnetic
multipole moments are negligible. The schematic of current geometry (red) and
the corresponding magnetic field (blue) are shown in the inset of Figure 3a.
The calculated electromagnetic near-field distributions at different cross
sections of the Si nanodisk are shown in Figure 3b and c which manifest
themselves as typical signatures of near-field excitation under the anapole
condition [6]. If the excitation source is switched to a tightly focused AP
beam, the Si nanodisk is turned into the resonant scattering condition, where
the dominant Cartesian electromagnetic multipole is electric quadrupole (EQ)
moment. The corresponding electromagnetic near-field distributions are shown
in Figure 3e and f, which validates qualitatively the dominant excitation of
an MQ resonance. It should be pointed out that there is still residual induced
MD moment accessed by the focused AP beam. More importantly, such high
contrast reconfigurable optical scattering is tunable by simply changing the
radius of the Si nanodisk, as shown in Figure 4a and b.As can be seen from
this figure, the MQ resonance overlaps with the anapole condition quite well
when the radius of the Si disk is changed. According to the calculated results
of the Si nanodisk with different radius under the excitation of AP and RP
beams, the wavelength of reconfigurable optical scattering can be effectively
tuned about 50 nm, indicating sufficient tolerance for experimental
demonstration.

Figure 5:  (a) A Si nanodisk fabricated on a glass substrate, where h = 160 nm
and r = 150 nm indicate its height and radius. SEM images of the Si nanodisk
from top (b) and side (c) views. (d) Optical setup for the back scattering
spectroscopy of CVBs. FSF, Fourier spatial filter; LCPC, liquid crystal
polarization converter; BS, beam splitter; CCD, charge coupled device; NA,
numerical aperture. (e) Experimental measured back scattering spectra of a Si
nanodisk excited by AP (blue) and RP (red) beams under the same excitation
power, respectively. The dashed lines are guide to the eye. CCD images under
the MQ resonance condition excited by a focused AP (f) and the anapole
condition excited by a focused RP (g). The excitation wavelengths for (f) and
(g) are both 735 nm.

##  3 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 Figure 3.
The reason for small scattering signal collected at the anapole condition
might be attributed to the morphology deformation (size difference and surface
roughness) in the fabricated sample utilizing mask based inductively coupled
plasma reactive ion etching (see supplement 1), as shown in the inset of Fig.
5 (b) and (c). In order to further access the scatteringless anapole
condition, the preparation of Si nanodisks with well-defined morphology is
subject to further optimization by e-beam lithography technology [48] and
colloidal synthesis technology [50]. Furthermore, we can confirm from the
dark-field back scattering images that the radiationless anapole condition is
approached under the RP excitation [see Figure 5g] while clear resonant
scattering can be visualized in Figure 5f. These experimental results validate
that the anapole condition can be realized without sophisticated tailoring of
electromagnetic multipole moments of nanoparticle. More importantly, one can
simply turn on or turn off the optical scattering of a fixed nanoparticle at
the same wavelength utilizing tightly focused AP and RP beams, validating the
concept of reconfigurable optical scattering for meta-optics. It is also
expected that such a mechanism can be applied in the nonlinear scattering
region, where distinct harmonic generation, stimulated Raman scattering and
saturated scattering can be manipulated by different cylindrical vector beam.

##  4 Conclusion

In summary, we propose a new mechanism to excite the radiationless anapole
condition without sophisticated manipulation of electromagnetic multipolar
moments of all orders to realize superpositions of vanished moment strengths
at the same wavelength. As a result, high contrast reconfigurable optical
scattering utilizing the unique combination of structured light and structured
Mie resonances can be realized. It means that a silicon nanoparticle whose
anapole condition is hidden in an electromagnetic multipolar resonance can be
selectively accessed as the radiationless state or resonant scattering state
by utilizing different tightly focused CVBs. More importantly, experimental
validation based on a simple Si nanodisk in the visible spectrum is provided
to further consolidate the proposed reconfigurable optical scattering. Our
results might provide a basic idea to realize a reconfigurable electromagnetic
atom for meta-optics. By combining both degrees of freedom in structured light
and structured Mie resonances, one might anticipate the possibility to realize
ultrafast manipulation of optical signal without applying optical nonlinear
and optomechanic effects.

## Acknowledgments

The authors acknowledge financial support from the National Key R&D Program of
China (YS2018YFB110012), National Natural Science Foundation of China (NSFC)
(Grant Nos. 11674130, 91750110, 61522504 and 61975067), Guangdong Provincial
Innovation and Entrepreneurship Project (Grant 2016ZT06D081), Natural Science
Foundation of Guangdong Province, China (Grant Nos. 2016A030306016,
2016TQ03X981 and 2016A030308010) and Pearl River Nova Program of Guangzhou
(No. 201806010040).

## Conflict of interest

The authors declare that they have no conflict of interest.

## Supplementary information

is available for this paper at https://doi.org/

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