Electric dipole-free interaction of visible light with silver metadimers P. Grahn, A. Shevchenko, and M. Kaivola Department of Applied Physics, Aalto University, P.O. Box 13500, FI-00076 Aalto, Finland (Dated: November 19, 2018) In subwavelength-sized particles, light-induced multipole moments of orders higher than the elec- tric dipole are usually negligibly small, which allows for the light-matter interaction to be accurately treated within the electric dipole approximation. In this work we show that in a specially designed meta-atom, a disc metadimer, the electric quadrupole and magnetic dipole can be the only excitable multipoles. This condition is achieved in a narrow but tunable spectral range of visible light both for individual metadimers and for a periodic array of such particles. The electromagnetic fields scat- tered by the metadimers fundamentally differ from those created by electric dipoles. A metamaterial composed of such metadimers will therefore exhibit unusual optical properties. In natural materials, effects of light-matter interac- used for creation of ultrathin optical elements, such as tion, such as optical refraction and reflection, are nearly beam splitters and phase retarders, with unusual optical completely governed by the electric dipole excitations in characteristics. the atoms and molecules composing the medium. While In this Letter, we propose a concept of metamaterial higher-order multipoles, such as electric quadrupoles and that interacts with light solely via electric quadrupole magnetic dipoles, can give rise to various optical phenom- and magnetic dipole excitations. These electromagnetic ena [1], their contribution to the overall interaction is multipoles are of the same order, and for brevity we refer very small. In optical metamaterials, however, light can to this type of material as quadrupole metamaterial. To excite also significant magnetic dipole moments, which facilitate a practical realization of this concept, we search can lead to such extraordinary optical phenomena as neg- for a unit structure for the material, a meta-atom, that is ative refraction and near-field focusing [2]. (i) simple, such that the material can be relatively easily Recently, it was shown that one can efficiently excite fabricated, (ii) much smaller than the visible-light wave- both magnetic dipole and electric quadrupole moments length, for the metamaterial to be treatable as homoge- at visible frequencies in a pair of silver bars [3]. In spite neous in the visible spectral range, and (iii) symmetric, of the fact that the bars had rather large dimensions, to enable a polarization independent response of the ma- the electric dipole contribution to the light scattering terial to the field at normal incidence. To reach this was still dominating in this structure. A metamaterial, goal, we employ a metadimer consisting of two closely in which the meta-atomic electric dipole moments would spaced metal nanoparticles. Such dimers have previously have a negligible contribution to the light scattering com- been studied for excitation of antisymmetric [6–9] or dark pared to the higher-order multipoles would be a unique [10–12] plasmon modes that are characterized by simul- optical material that has been neither observed in nature taneous oscillation of two oppositely directed electric cur- nor studied in scientific experiments. The radiation pat- rents. The currents in those dimers, however, were only terns of electric quadrupoles and magnetic dipoles funda- partially opposite and did not have equal effective am- mentally differ from those of electric dipoles. Therefore, plitudes. As a result, the dominating scattering by the light propagation phenomena in such higher-order meta- structures still originated from electric dipoles. In order materials must be treated in a conceptually different way to exactly identify and evaluate the light-induced multi- than in an ordinary medium. As an example, according pole moments in our meta-atoms, we use the electromag- to the recently introduced electromagnetic boundary con- netic multipole expansion introduced, e.g., in Ref. [13]. ditions in the presence of electric quadrupoles, the tan- The metadimer we propose as a unit of a quadrupole arXiv:1204.0618v1 [physics.optics] 3 Apr 2012 gential components of the electric and magnetic fields can metamaterial consists of two axis-aligned silver discs, be discontinuous across the material boundaries [4, 5]. arranged as depicted in Fig. 1. The discs have the Furthermore, if the electric polarization P in the mate- same thickness h1 = h2 = 10 nm, but different radii, rial could be made negligible, the interaction of light with R1 = 15 nm and R2 = 20 nm. The surface-to-surface the material would be described by Maxwell’s equations separation of the discs is set to s = 10 nm. In our cal- in which P is absent. Such a material can exhibit reduced culations, we choose the coordinate system to have the optical reflection and refraction at the interface with vac- origin located on the dimer axis, at the center of the gap uum. If, in addition, the absorption could be made low, between the discs. The z-axis is directed such that the the material itself could become essentially invisible. It smaller disc is in the z < 0 half-space, while the larger is also remarkable that, in contrast to electric dipoles, one is in the z > 0 half-space. In order to have a scenario the fields radiated by light-induced electric quadrupoles that can be realized in practice, the discs are considered and magnetic dipoles in the forward and backward di- to be embedded in a homogeneous dielectric of refractive rections oscillate out of phase. This property could be index 1.5. This choice implies no loss of generality and 2 the conclusions to be drawn are equally applicable to any (a) x10 -14 other choice of lossless surrounding. 1.5 ] 2 1 R [m 1 e s 0.5 C h 0 1 400 450 500 550 600 650 700 λ [nm] s 0 e m q R2 -17 C (b) x10 Cs Cs s 6 h2 ] 2 4 [m s C 2 FIG. 1. Illustration of the metadimer geometry. 0 580 585 590 595 600 605 610 We assume that the metadimer is illuminated by a lin- λ 0 [nm] early x-polarized plane wave propagating in the +ˆz di- rection. Because of the chosen size and geometry for FIG. 2. Spectra of the modal scattering cross sections of a sil- the metadimer, all moments of higher order than the ver metadimer. The metadimer is embedded in glass and has electric quadrupole and magnetic dipole moments can the dimensions (R1,h1,R2,h2,s) = (15,10,20,10,10), in nm. (a) e be neglected. The symmetry of the metadimer with re- The whole visible spectrum of the cross section Cs due to the spect to the illumination direction and polarization en- electric dipole mode. (b) The cross sections originating from sures that the electric dipole, magnetic dipole and elec- each of the three lowest order modes around the wavelength of the electric dipole suppression. tric quadrupole moments are p = ˆxp , m = ˆym , and ←→ x y q = (ˆxˆz + ˆzˆx)qxz, respectively. Here ˆxˆz and ˆzˆx are the outer products of the unit vectors ˆx and ˆz. Fig. 3. In the spectral region around 594 nm, the electric We first calculate the scattered electromagnetic field dipole moments of the two discs oscillate out of phase distribution around the metadimer using the computer with respect to each other (see Fig. 3a). A complete software COMSOL Multiphysics. The values of the elec- electric dipole suppression is obtained if, in addition, the tric permittivity of silver are taken from Ref. [14]. We two dipole moments have equal amplitudes. Figure 3b then expand the scattered field by using the multipole ex- shows that the magnitudes of the dipole moments are pansion [13, 15]. The expansion coefficients are used to e m q indeed equal at the wavelength of 594 nm, which is in calculate the contributions Cs , Cs and Cs of the electric agreement with Fig. 2. dipole, magnetic dipole, and electric quadrupole, respec- We have verified by calculations that increasing the tively, to the scattering cross section of the metadimer. separation between the two discs will blue-shift the spec- These modal cross sections describe the amount of opti- tral location of the electric dipole suppression. For exam- cal power radiated to the far-field by the corresponding ple, at a separation of s = 30 nm this location is shifted multipole moment, relative to the intensity of the inci- to a vacuum wavelength of 546 nm. Thus, varying the dent plane wave. Thus, the modal cross sections enable separation allows us to tune the spectral location of the us to compare the different multipole excitations. dominating electric quadrupole and magnetic dipole scat- The modal cross sections of the metadimer as a func- tering. tion of the vacuum wavelength λ0 are depicted in Fig. 2. Next, we show that in the multipole expansion of the Figure 2a shows the electric dipole contribution to the electric current in the metadimer, the electric quadrupole scattering cross section over the whole visible spectrum. and the magnetic dipole are interconnected excitations of It can be seen that at around λ0 = 594 nm, the electric the same order. The small size of the metadimer allows dipole moment is suppressed and, as Fig. 2b indicates, us to treat it as a point particle. Neglecting all multipole the power scattered by the electric dipole mode is neg- moments of higher order than the electric quadrupole and ligibly small compared to that scattered by the electric magnetic dipole, the electric current density of a point quadrupole and magnetic dipole modes. While the two particle can be written in the form (see, e.g., [16]) remaining cross-sections Cm and Cq are small, they en- s s ←→ tirely determine the light-metadimer interaction at this J(r)= −iωp − q · ∇δ(r) − m × ∇δ(r).