Nonradiating anapole modes in dielectric nanoparticles

Abstract

Nonradiating current configurations attract attention of physicists for many years as possible models of stable atoms. One intriguing example of such a nonradiating source is known as 'anapole'. An anapole mode can be viewed as a composition of electric and toroidal dipole moments, resulting in destructive interference of the radiation fields due to similarity of their far-field scattering patterns. Here we demonstrate experimentally that dielectric nanoparticles can exhibit a radiationless anapole mode in visible. We achieve the spectral overlap of the toroidal and electric dipole modes through a geometry tuning, and observe a highly pronounced dip in the far-field scattering accompanied by the specific near-field distribution associated with the anapole mode. The anapole physics provides a unique playground for the study of electromagnetic properties of nontrivial excitations of complex fields, reciprocity violation and Aharonov-Bohm like phenomena at optical frequencies.

ARTICLE

Received 10 Jan 2015 | Accepted 15 Jul 2015 | Published 27 Aug 2015

Andrey E. Miroshnichenko1, Andrey B. Evlyukhin2,3, Ye Feng Yu4, Reuben M. Bakker4, Arkadi Chipouline5, Arseniy I. Kuznetsov4, Boris Lukyanchuk4, Boris N. Chichkov2,3,6 & Yuri S. Kivshar1

Nonradiating current congurations attract attention of physicists for many years as possible models of stable atoms. One intriguing example of such a nonradiating source is known as anapole. An anapole mode can be viewed as a composition of electric and toroidal dipole moments, resulting in destructive interference of the radiation elds due to similarity of their far-eld scattering patterns. Here we demonstrate experimentally that dielectric nanoparticles can exhibit a radiationless anapole mode in visible. We achieve the spectral overlap of the toroidal and electric dipole modes through a geometry tuning, and observe a highly pronounced dip in the far-eld scattering accompanied by the specic near-eld distribution associated with the anapole mode. The anapole physics provides a unique playground for the study of electromagnetic properties of nontrivial excitations of complex elds, reciprocity violation and AharonovBohm like phenomena at optical frequencies.

DOI: 10.1038/ncomms9069 OPEN

Nonradiating anapole modes in dielectric nanoparticles

1 Nonlinear Physics Centre, The Australian National University, Acton, Australian Capital Territory 2601, Australia. 2 Laser Zentrum Hannover e.V., Hollerithallee 8, D-30419 Hannover, Germany. 3 Institute of Laser and Information Technologies RAS, 142190 Moscow, Troitsk, Russia. 4 Data Storage Institute, A*STAR, 5 Engineering Drive 1, 117608, Singapore. 5 Technische Universitat Darmstadt, Institut fr Mikrowellentechnik und Photonik, Merckstra e 25, 64283 Darmstadt. 6 Saint-Petersburg Polytechnic University, Polytechnicheskaya 29, 195251 St Petersburg, Russia. Correspondence and requests for materials should be addressed to A.E.M. (email: mailto:[email protected]

Web End [email protected] ).

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The possible existence of non-radiating sources has puzzled physicists since the early days of electromagnetic theory, particularly in connection with models of stable atoms

and electrons congurations13. The term anapole (that means without poles in Greek) was introduced in the physics of elementary particles by Yakov Zeldovich4. Recently, an anapole was suggested as a classical model of elementary particles describing dark matter in the Universe5. The electrodynamics analogue of the anapole is a composition of electric and toroid dipole moments, without far-eld radiation due to complete destructive interference of their similar radiation patterns6. The toroidal modes and an analogue dynamic anapoles have been demonstrated in the microwave part of the spectrum7, toroidal resonances were also seen at optical frequencies8. The classical analogue of a stationary anapole is the well-known toroid with a constant poloidal surface current4. This current distribution is also associated with a toroidal dipole moment pointing outward along the torus symmetry axis (see Fig. 1). The static magnetic eld produced by a toroid is entirely concentrated within a coil in the form of a circulating magnetic current (see Fig. 1). It generates no eld outside, but may possess a non-zero potential, which might lead to violation of the reciprocity theorem and Aharonov Bohm like phenomena9,10.

In general, the existence of toroidal multipoles is required by the symmetry of the order parameters with respect to the inversions of space and time: in addition to the well-known electric polarization (described by an r-odd and t-even polar vector) and magnetization (r-even and t-odd axial vector), there should be the order parameters described by an r- and t-even polar vector and an r- and t-odd axial vector11. The latter two correspond to the toroidal ordering and have been observed in natural crystals (see, for example, ref. 12, for a recent review) as well as in articial metamaterials13. A radiationless nuclear anapole moment has been observed in atomic Cesium14 through atomic parity-violating effects, which were rst suggested in ref. 15. In the Standard Electroweak theory, nuclear anapole moments arise due to parity-violating forces inside the nucleus. However, the interactions of nucleons with a static cosmic eld and an axion dark matter eld can also give rise to static and oscillating nuclear anapole moments, respectively, (see for example, refs 16,17).

In the dynamic case, an oscillating toroidal dipole moment produces non-zero electromagnetic radiation with a pattern fully repeating that of an electric dipole moment but being scaled by a factor o2, where o is the angular frequency of light (see Fig. 1).

For the oscillating surface current, its radiationless properties can be realized by exciting a second electric dipole oscillating out-of-phase with the toroid, resulting in complete destructive interference of their radiation due to equivalence of their scattering patterns. Such a radiationless nontrivial oscillating current conguration was (in analogy with non-oscillating

poloidal current) also named anapole6. However, this type of the radiation compensation, in general, is not complete for an external excitation: the compensated toroidal dipole moment is a part of the third-order source current multipole expansion. Other terms of the same order (magnetic quadrupole and electric octupole) can remain radiative. Nontrivial nonradiative current congurations have attracted continuous interest since the early days of electrodynamics; from the fundamental physics to applications of nonscattering objects18,19. Since ideal nonscattering objects do not exist20, by radiationless we refer to a compensation or absence due to structural peculiarities of the leading multipole order; below we term the respective combination as the anapole mode.

The concept of toroidal modes have attracted considerable attention in the eld of metamaterials as a possible realization of radiationless objects7,2125. The toroidal moment itself and its associated effects (including toroidal metamaterials13,25) have been recently studied theoretically8,22,23,2629. Several experimental verications in microwave7,13,21 and optical30 domains for toroidal moments have conrmed the proposed theory. Moreover, the mutual compensation of the toroidal and dipole moments has been experimentally observed in GHz frequency region with a specially designed structure7.

Here we demonstrate experimentally, for the rst time to our knowledge, the existence of an anapole mode in optics in the simple structure of a silicon nanodisk. We observe a strong suppression of the far-eld scattering along with nontrivial evolution of the electromagnetic eld inside a nanodisk close to the wavelength of the anapole mode excitation. Very recently, excitation of a toroidal moment at THz frequencies has been theoretically predicted for quadrumers of dielectric cylinders made of LiTaO3 (ref. 8).

ResultsTo analyze the electromagnetic properties of the silicon nanoparticle theoretically and to demonstrate the anapole excitation in the system, we employ multipole expansions in the two representations: for elds outside the nanoparticle in the spherical multipoles31 and in the Cartesian for currents inside the nanoparticle; each series can be unambiguously expressed one through the other30. For the sake of generality, we assume a nontrivial current distribution J(r, t) producing an electromagnetic eld E(r, t).

We employ the eld multipole expansion, which, due to orthogonally of vector spherical harmonics, allows us to unambiguously study the radiation properties by representing the total scattering cross-section as a sum of intensities of spherical electric aE(l, m) and magnetic aM(l, m) scattering coefcients32:

Csca

p k2

X1 l1

Xlm l 2l 1 aE l; m j j2 aM l; m j j2

1

where the scattering coefcients

aE l; m

i

l 1kZE0

Electric dipole Toroidal dipole Anapole

Z Y l;my; fjlkr

k2r J r

2 r

+

d3r

2

p

p 2l 1

l 1

d dr

= Jr

aM l; m

i

l 1k2ZE0

p

Z Y lmy; jjlkrr

Figure 1 | Illustration of an anapole excitation. The toroidal dipole moment is associated with the circulating magnetic eld M accompanied by electric poloidal current distribution. Since the symmetry of the radiation patterns of the electric P and toroidal T dipole modes are similar, they can destructively interfere leading to total scattering cancelation in the far-eld with non-zero near-eld excitation.

p 2l 1

l 1

= Jr

d3r 3

can be calculated directly through the induced currents inside the structure31. The coefcients aE(l, m) and aM(l, m) are called eld

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spherical multipole coefcients32 (see also Supplementary Methods).

At the same time, the source current distribution can be expanded over current Cartesian multipole coefcients26,33,34:

J r; t X

l0

1 l

l ! Bli:::k@i:::@kd r

R

E

Bli:::k Z J r; tri:::rkd3r

2 j j2dV integrated over the spherical volume normalized to

the trivial case of a transparent particle Win 2pR3|E0|2/3. It

again demonstrates that at the condition of zero scattering the energy density is non-zero and larger than in trivial caseit means that the energy is concentrated inside the particle.

To analyse the excitation of the anapole mode in more details, we consider the properties of a single-spherical electric dipole. In this case, the response of the electric dipole mode can be also described in terms of effective dipole moments. The contribution to the far-eld scattering with the electric dipole symmetry can be written as

Esca

4

Here Bli:::kis a tensor of rank l. We would like to note that the structure and interpretation of these coefcients requires a special care. In particular, from these tensors, various current Cartesian multipoles can be obtained. For example, B1i d1idetermines the electric dipole moment d1i, B2ij Q2ij m1i consists of electric quadrupole Q2ij(symmetric) and magnetic dipole m1i (anti-symmetric) moments, and B3ijk O3ijk m2ij T1i gives rise to electric octupole O3ijk, magnetic quadrupole m2ij, and toroidal dipole moments T1i. Note that the toroidal dipole moment of the current appears in the third-order expansion coefcients33, in contrast to the eld expansion, and is associated with a particular current congurations. Field expansion and respective eld spherical multipole coefcients aE(l, m) and

aM(l, m) are not associated with a particular current, but rather with the scattered eld structure. For example, in the far eld, the dipole and toroidal dipole radiation patterns are indistinguishable and cannot be recognized in the frame of the eld spherical multipole expansion. From the other perspective, current Cartesian multipole expansion allows us to unambiguously separate the dipole and toroidal dipole contributions, and, hence, provides with a tool necessary for the required structure design. The current expansion can also be performed over any full orthogonal basis, for example over vector spherical harmonics9, where the contribution of the toroidal moments and corresponding current congurations can be uniquely identied.

We start our analysis from a simple observation of some peculiar properties of light scattering by spherical dielectric particles. In the Mie theory, the transverse boundary conditions can be written independently for each multipole order as ref. 35

cl nx

dE l; 1

xl x

aE l; 1

cl x

;

c0l nx

dE l; 1

nx0l x

aE l; 1

k24pe0 n Psph n; Psph 6ipe0k3 aE1; 1 6

where n is the normalized vector of observation direction and the effective electric dipole moment is dened through the corresponding Mie scattering coefcient. This is a description of the radiation properties of an electric dipole by using a eld spherical multipole coefcient.

However, electric dipole moments inside the particle can be obtained via integrating the induced current over the whole volume

Pcar

i o

Z Jdr 7

with

J ioe0 n2 1

E

8

This is a description of the electric dipole moments inside the particle by using the current Cartesian multipoles. In Fig. 2c we plot the dependence of both electric dipole moments for a dielectric sphere versus diameter, where only the electric eld of the spherical electric dipole inside the particle was taken into account. This gure demonstrates that for small particle sizes both approaches produce similar results. For larger particles, the two descriptions deviate from each other quite strongly. In particular, there is a situation for the diameter B204 nm, when the electric dipole scattering vanishes, Psph 0, while the induced electric

dipole inside the particle is non-zero, |Pcar|a0. In this case we

have an apparent contradiction that there is a source of the electric dipole response, but no contribution to the far-eld. Similarly, an opposite situation exists for the diameter around 196 nm, when the induced electric dipole moments inside the particle vanishes, Pcar 0, while there is a non-zero contribution to the far-eld

|Psph|a0. In this case, we have a nite contribution to the far-eld in the absence of a source! To resolve such contradictions, we note that for larger diameters, the electric eld distribution inside the particle becomes highly inhomogeneous; simple averaging of the induced current (proportional to the electric eld) is not enough to properly describe the contribution to the far-eld. As was mentioned above, in general, the scattering coefcients can be expressed through Cartesian multipole coefcients Bli:::k. The relation between current Cartesian and eld spherical multipoles for the electric dipole coefcient aE can be written as ref. 32

aE1; 1 C1 B1x iB1y

h i

7C3 B3xxx 2B3xyy 2B3xzz B3yyx B3zzx

h i

i B3yyy 2B3yxx 2B3yzz B3xxy B3zzy

h i

5

where x 2pR/l is the size parameter, n is the relative refractive

index of the particle, clx and xlx are RiccatiBessel functions

and dE(l, 1) are the internal coefcients of electric multipole excitation inside the spherical particle31. In the limit of a transparent particle (n-1), all the scattering coefcients vanish aE(l, 1) 0 and the amplitude of the internal coefcients tend to

unity, |dE(l, 1)| 1. This is a trivial case of a nonscattering particle

without induced polarization inside. In the nontrivial case with n41, in general, there are conditions when aE(l, 1) 0

and |dE(l, 1)|41, corresponding to the situations with non-zero induced polarization inside the particle and, at the same time, zero scattering of a particular multipolar order. This condition exhibits the existence of non-radiative sources in the simplest spherical geometry. Since, for each multipole there exist an innite number of such conditions, they provide a recipe to create a variety of nonradiative current congurations. In Fig. 2a we show the dependence of electric dipole scattering |aE(1, 1)| and internal |dE(1, 1)| coefcients of a dielectric spherical particle as a function of diameter for refractive index n 4 and wavelength

550 nm, which demonstrates the situation of a vanishing scattering coefcient aE(1, 1) 0 associated with the non-zero

induced eld. At the same time, in Fig. 2b we present the partial scattering contribution of the electric dipole Cscap|aE(1, 1)|2

together with the corresponding electric energy WE

n2

nc0l x

;

9

where C1 ik3/(6pe0E0) and C3 ik5/(210pe0E0) (ref. 30).

We notice, that among these higher order current Cartesian coefcients there are toroidal dipole moments T1iwith a radiation

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a b

Scattering

Electric energy

Mie coefficients

43.5 3

2.5 2

1.5 1

0.5 0

Partial scatterieng (nm2 )

0.140.120.10.080.060.040.02 0

Normalized electric energy

120

140

160

180

200

220

240

120

140

160

180

200

220

240

Diameter (nm)

|a (1,1)|

|d (1,1)|

Diameter (nm)

14 12 10 8 6 4 2 0

c

14

12

10

8

6

4

2

0

Spherical: Electric dipole Cartesian: Electric dipole

Toroidal dipole

Scattered electric field, E S(a.u.)

Anapole

Toroid only

Electric dipole

120

130

140

150

160

170

180

190

200

210

220

230

240

Diameter (nm)

Figure 2 | Decomposition of the scattering in Spherical and Cartesian multipoles. We consider scattering by a dielectric spherical particle inside as a function of diameter for refractive index n 4 and wavelength 550 nm: (a) Scattering |aE(1, 1)| and internal |dE(1, 1) Mie coefcients; (b) partial scattering

cross-section and energy density of the electric dipole; (c) calculated spherical electric dipole |Psph|(black), Cartesian electric |Pcar| (red) and toroidal |Tcar| (green) dipole moments contributions to the partial scattering. These gures demonstrate that for small particles both contributions of the spherical and

Cartesian electric dipoles are identical and the toroidal moment is negligible. For larger sizes, the contribution of the toroidal dipole moments to the total scattered eld has to be taken into account. The anapole excitation is associated with the vanishing of the spherical electric dipole Psph 0, when the

Cartesian electric and toroidal dipoles cancel each other.

pattern similar to the electric dipole (see, for example, Supplementary Information in ref. 33). It implies that it now should be taken into account to correctly describe the scattering properties. The toroidal dipole moment can be calculated via induced current as

Tcar

1 10c

Z r J

r 2r2J dr

10

and the total contribution to the far-eld scattering can now be written as (see Supplementary Methods)

Esca

k24pe0 n Pcar n ikn Tcar n

f g: 11

Thus, the far-eld radiation vanishes, Esca 0, when the

contributions of the current Cartesian electric and toroidal dipoles to the scattered eld are out-of-phase

Pcar ikTcar 12 and, thus, interfere destructively with each other. This is the exact condition for excitation of the radiationless anapole mode.

In Fig. 2c we also show the dependence of the toroidal dipole contribution to the far-eld, which becomes essential for larger sized particles. In particular, when the induced electric dipole moment vanishes, Pcar 0, only the toroidal dipole contributes to

the far-eld scattering. Alternatively, the scattering cancellation is possible, Psph 0, when non-zero electric and toroidal dipoles

contributions compensate each other in the far-eld. Thus, the partial scattering can vanish meaning that the electric dipole scattering coefcient becomes zero, aE(1,1)E0, which can be achieved when the rst-order (electric dipole) coefcients B1i have to be compensated by the toroidal part of the third-order coefcients B3ijk. This simple consideration creates the basis for understanding of physics of the anapole mode, namely, mutual compensation of lower B1iand part of the higher B3ijkorder components with a nontrivial current distribution inside the particle.

All of this raises an important question: is it possible to experimentally observe the anapole mode? By using external sources it is not possible to achieve total zero scattering due to Lorentz reciprocity. But, can we get around this condition with

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a

Anapole field distribution

Electric field on top

Electric Magnetic

0.50.40.30.20.10.0

c d

Diameter (nm)

350 300 250 200

x

500

600

700

800

Wavelength (nm)

2.52.01.51.00.5 0

Scattering cross-section

Multipoles decomposition, D=310 nm

b e f

Spherical Cartesian

0.50.40.30.20.10.0

Total scattering Electric dipole Toroidal dipole Magnetic dipole Magnetic quarupole

Diameter (nm)

350 300 250 200

Scattering cross-section (m2 )

Scattering cross-section (m2 )

Total scattering Electric dipole Magnetic dipole Magnetic quadrupole

500

600

700

800

550

Wavelength (nm)

600

650

700

750

800

550

Wavelength (nm)

600

650

700

750

800

Wavelength (nm)

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0

Figure 3 | Numerical results for a single silicon nanodisk under normal incidence. (a) Electric eld on top of the nanodisk of thickness h 50 nm at the

surface centre for various diameter. (b) Total scattering cross-section spectra of the silicon nanodisks with similar parameters. The scattering suppression is accompanied by the electric eld enhancement in the centre of the nanodisk. (c,d) Electric and magnetic scattered near-eld distribution at the anapole wavelength corresponding to the mimimum of the far-eld scattering. (e,f) Spherical and Cartesian multipole decompositions of the scattering spectra for diameter 310 nm. In eld spherical harmonics only the electric dipole is dominant, which becomes suppressed at 650 nm wavelength. On contrary, current Cartesian multipoles exhibit two leading contributions from electric Pcar and torodial Tcar dipoles, which are out ofphase and, thus, compensate each other in the far-eld.

realistic structures? The spherical geometry under planewave illumination is not suitable, since in the vicinity of the anapole excitation other multipoles will produce strong contributions to the total scattering due to superposition. For successful experimental observation of the anapole we need to nd a structure, which exhibits leading contribution of solely the electric dipole with all other modes being strongly suppressed. One of the possibilities to achieve this is to use a disk geometry with a particular aspect ratio. Specically, we found that a silicon nanodisk of height 50 nm and diameter ranging from 200 to 400 nm supports the anapole mode, which can be experimentally excited and observed (see Fig. 3). For this geometry, calculations are made in CST Microwave Studio show a strong dip in the far-eld scattering spectrum (see Fig. 3b), accompanied by a near-eld enhancement inside and around the disk (see Fig. 3a), indicating the presence of the combination of electric and toroidal dipoles which provide the anapole mode conguration. Opposite circular displacement currents in the left and right hand sides of the disk (see Fig. 3c) generate a circular magnetic moment distribution that is perpendicular to the disk surface (see Fig. 3d). This provides a strong toroidal moment oriented parallel to the disk surface.

Excitation of this toroidal dipole and its destructive interference with the electric dipole is conrmed through current Cartesian and eld spherical multipole decompositions (see the 310-nm diameter disk in Fig. 3e,f). One advantage of such disks compared with other geometries, for example, spheres, is that the other multipoles apart from the electric and toroidal dipoles are suppressed (see also Supplementary Fig. 1 for a silicon disk with diameter of 200 nm). We clearly observe a strong suppression of the total scattering due to the far-eld cancelation of electric and toroidal dipole radiations (see Fig. 3f), which is exactly the condition for anapole mode excitation. Since we are exciting the

silicon disk with an incident plane wave, the total scattering cannot completely vanish due to reciprocity. Instead, magnetic quadrupole radiation is dominant in the far-eld (see Fig. 3e). At the same time, the near-eld distribution is quite complex and does not correspond to a magnetic quadrupole only. The multipole expansion of current distribution inside the silicon nanodisks (see Supplementary Methods) indicates dominant contributions of electric and toroidal dipole moments (see Fig. 3f). The presence of magnetic quadrupole in disk geometry can be also understood from the structure of the toroidal dipole moment, schematically presented in Fig. 1. In particular, for an ideal spherical geometry, a toroidal dipole is characterized by the presence of the circulating magnetic ux along the closed loop. By transforming a sphere to a disk, this loop discontinues and leads to the formation of two anti-parallel magnetic moments, resulting in magnetic quadrupole radiation, which does not affect the condition of the anapole mode excitation. Our direct numerical results indicate that the anapole excitation in silicon disks is robust against incident angle and polarization (see also Supplementary Fig. 2).

To conrm these theoretical predictions, we performed a series of experiments to observe the anapole excitation. Silicon nanodisks with a height of 50 nm and diameters ranging from 160 to 310 nm were fabricated on a quartz substrate using standard nanofabrication techniques (see Methods). Far-eld scattering spectra (Fig. 4a) were measured using single nanoparticle dark-eld spectroscopy (see Methods). For disks with a diameter 4200 nm, a scattering dip appears around 550 nm; as the diameter increases, the dip redshifts and becomes more pronounced. The spectral position of this far-eld scattering minimum is in good agreement with the theoretically predicted anapole excitation (see Fig. 3b for comparison). To show that the spectral dip in far-eld scattering corresponds to the dark anapole

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a b

Scatetering spectra

4,000

SEM image

NSOM setup

Dark field scattering (a. u.)

3,000

285 nm

260 nm

2,000

235 nm

210 nm

310 nm

1,000

185 nm

160 nm

0

500 600 700 800

Wavelength (nm)

c d

610 nm

620 nm

630 nm 640 nm 650 nm 660 nm

2.5

E

21.5 10.5 0

0 3.5 32.5 21.5 10.5 0

Measured field,

a.u.

Electric field,

Et / E0

Magnetic field,

Ht / H0

300 nm

2

1.5

1

0.5

Et

Ht

Disk diameter - 310 nm

Figure 4 | Experimental demonstration of the anapole mode in a silicon nanodisk. (a) Experimental dark eld scattering spectra of silicon nanodisks with a height of 50 nm and a diameter ranging from 160 to 310 nm. The baseline for each spectrum has an offset step of 500 A.U. (b) SEM imagesof the two largest disks (for viewing convenience the view is tilted at 52). The scale bar, 200 nm. (c) Near-eld enhancement around the silicon nanodisk with diameter of 310 nm; the top row shows experimental NSOM measurements while the middle and bottom rows show calculated transversal electric and magnetic near-eld, respectively, on top of the disk, 10 nm above the disk surface. White dashed lines in the experimental images indicate the disk position. Polarization of the excitation light is shown in the gure. (d) Schematic representation of NSOM measurement with the incident light coming through the substrate and collecting on the top with the metal-coated tip.

mode excitation, the near-eld distribution around the disks is mapped at multiple wavelengths using a near-eld scanning optical microscope (NSOM) and a supercontinuum light source (see Methods). The observed deviation from the theoretically predicted suppression is primarily attributed to the non-perfect disk shape and inevitable roughness. Representative experimental near-eld maps for the 310-nm diameter disk at wavelengths around the anapole excitation are provided in Fig. 4b, compared directly with simulated near-eld maps of transverse electric and magnetic eld at a height of 10 nm above the nanodisk. The numerical simulations were performed by using CST Microwave Studio for a bar silicon nanodisk and the near-eld was collected above the nanoparticle for linear polarized planewave excitation coming from the bottom. In the simulations we did not take the presence of the probe into account. It is known, that the signal collected by an aperture-type near-eld probe is not directly proportional to any of the near-eld components separately but is a result of a convolution integral of both electric and magnetic near-elds (see for example, ref. 36). Thus, it should reect near-eld maxima of both electric and magnetic eld components. For this reason in Fig. 4 we compare simulated electric and magnetic near-elds with the experimental NSOM data. The experimental near-eld maps show the evolution of the spectral response throughout the visible. As the wavelength approaches 620 nm, we begin to see the splitting of the central hotspot into two separate spots. Close to the anapole mode wavelength, at 640 nm a new hotspot appears in the middle of the disk and its intensity increases with wavelength. Experimental results clearly show the

appearance of a near-eld maximum at the anapole wavelength in the middle of the disk, which corresponds to appearance of the maximum of electric near-eld in simulations. The experimental results also show particular symmetry of the near-eld excitation with main lobes aligned perpendicular to polarization direction. This is also in a good correlation with both electric and magnetic near-eld components in the simulations. Similar near-eld behaviour with an anapole excitation at 620 nm is observed for a disk with diameter of 285 nm (see Supplementary Fig. 3).

In summary, the anapole mode observed in the silicon nanodisk originates from the interference of two different dipole moments. Such radiationless excitation can make the nanodisk almost invisible in the far-eld at the anapoles excitation wavelength. The anapole mode offers a new way to achieve an invisibility condition for lossless dielectric nanostructures based on the cancellation of radiation scattering (originally proposed by Kerker in (ref. 37) for metallic core-shell nanoparticles). Moreover, the relation between the electric and toroidal dipoles, including their mutual cancelation in the far-eld can be directly extended to magnetic moments, when the magnetic dipole radiation vanishes. Recent observations of a strong magnetic dipole response from silicon nanoparticles3840 suggests that it could be an ideal platform for the demonstration of such magnetic type anapoles.

We also mention that the anapole mode is not only limited to disk geometries but can also be observed for spheres38 or other dielectric nanostructures where the electric dipole contribution vanishes due to excitation of other modes with electric dipole

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symmetry in the far-eld. For other geometries, however, this effect can be hidden by contributions of higher-order multipole modes (quadrupoles, octupoles and so on). Similar effects can also be expected in metallic systems supporting higher-order dipole modes.

Methods

Fabrication. Silicon nanodisks with various diameters were fabricated on quartz substrates by standard nanofabrication techniques. A 50-nm silicon lm was grown on a quartz substrate using chemical vapor deposition (ICP-CVD, Oxford Instruments). A thin (o60 nm) layer of negative resist (HSQ) was coated on the sample. Lithography was performed with an electron beam lithography machine (Elionix 100 KV). After electron beam exposure and development, reactive ion etching (ICP, Oxford Instruments) was used to transfer the pattern into the silicon lm. The result is silicon disks with a thin (o10 nm) cap of residual resist on top of a quartz substrate.

Far-eld spectroscopy. Spectral analysis was performed using a single-nano-particle spectroscopy setup with a dark-eld geometry (see ref. 40 for details). The sample was irradiated by a halogen lamp source at an angle of 58.5. According to our simulations the spectral position of the anapole excitation practically does not change for variations of the incidence angle in the range from 0 to 60 (see Supplementary Fig. S3). Scattered light is collected from above into a solid angle corresponding to the microscope objective lens with 0.55 numerical aperature. The collected scattering spectra were normalized to the halogen lamp spectrum measured in a bright-eld reection geometry.

Near-eld. The sample was characterized in the near-eld using a NSOM (Multiprobe SPM/NSOM Cryoview, Nanonics Imaging). The sample was illuminated from the far-eld and the near-eld was collected through a subwavelength aperture (50-nm aperture of a tapered bre, coated with Chrome and Gold) in a transmission conguration. The light source used is a super-continuum source (SuperK Power, NKT Photonics). Specic wavelengths were selected using a variable bandpass lter (SuperK Varia, NKT Photonics). Photons were counted with Avalanche Photo Diodes (Excelitas Technologies). Scans on individual particles were performed with various wavelengths over a 2 2 mm

area with a pixel size of 8 nm. The near-eld maps are normalized, taking the

background as unity.

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Acknowledgements

We thank N. Zheludev and M. Berry for useful discussions, and their great interest to this

work. The work of A.E.M. was supported by the Australian Research Council via Future

Fellowship program (FT110100037). The authors at DSI were supported by DSI core

funds. Fabrication, Scanning Electron Microscope Imaging and NSOM works were

carried out in facilities provided by SnFPC@DSI (SERC Grant 092 160 0139). Zhen Ying

Pan (DSI) is acknowledged for SEM imaging. Yi Zhou (DSI) is acknowledged for silicon

lm growth. Leonard Gonzaga (DSI), Yeow Teck Toh (DSI) and Doris Ng (DSI) are

acknowledged for development of the silicon nanofabrication procedure. B.N.C.

acknowledges support from the Government of Russian Federation, Megagrant No.

14.B25.31.0019.

Author contributions

A.E.M., A.B.E. and A.C. developed the original concept and theoretical description.

Y.F.Y., R.M.B. and A.I.K. designed experiments and performed nanofabrication and

optical near- and far-eld measurements. All authors contributed to the writing of the

manuscript. A.E.M. supervised the project, analysed the data and coordinated the

manuscript preparation.

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modes in dielectric nanoparticles. Nat. Commun. 6:8069 doi: 10.1038/ncomms9069

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