Electric Dipole-Free Interaction of Visible Light with Silver Metadimers

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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 direction. 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). (1) wavelength. To obtain an intuitive picture of the electric current At λ0 = 594 nm, the dominating excitation in the distribution in the metadimer, we calculate the electric metadimer can be seen as a second order electric current dipole moments excited in each of the coupled discs. The excitation, a current quadrupole. The major contribu- magnitudes and phases of these moments are depicted in tion to the excited current quadrupole can be considered 3

(a) to assume the form [17] 450 Disc (z < 0) 3 ikr 360 Disc (z > 0) q IxzLs k e E (r)= cos θθˆcos θ cos φ − φˆsin φ. 270 Diﬀerence F ω 4πǫ r ) [deg] 0 (4) E

/ 180 x In contrast to an electric dipole, which radiates symmet- p 90 rically around its axis, the current quadrupole radiates 0 primarily in the +ˆz and −ˆz directions. Furthermore, the phase( -90 fields radiated in the +ˆz and −ˆz directions oscillate out 500 550 600 650 700 λ of phase. The numerically calculated far-field radiation 0 [nm] (b) x10 -32 pattern of the metadimer, at the wavelength of the electric dipole suppression, is depicted in Fig. 4a. For com- Disc (z < 0) parison, Fig. 4b shows the far-field radiation pattern of

/V] 2 2 Disc (z > 0) an electric dipole. The ﬁeld proﬁle in Fig. 4a is in good agreement with Eq. (4).

| [Cm 1 0 E / x p | 0 500 550 600 650 700 λ 0 [nm]

FIG. 3. Spectra of (a) the phase and (b) the absolute value of the x-components of the excited dipole moments in the individual discs of the metadimer, normalized to the electric ﬁeld amplitude E0 of the incident light.

to consist of two opposite x-oriented current elements of FIG. 4. (a) Far-field radiation pattern of the metadimer at length L separated by a distance s along the z-axis. If the wavelength of 594 nm. The excitation field is x-polarized. the element located at z > 0 carries a complex-valued (b) Far-field radiation pattern of an x-oriented electric dipole. Arrows: electric field vectors. Colors: normalized intensity. current of +Ixz and the one at z < 0 a current of −Ixz, the electric current density in the point particle limit is [17] To create a quadrupole metamaterial that can be treated as homogeneous, the metadimers should be ar- d ranged in a lattice with a short period Λ << λ0. Thus, Jxz(r)= −ˆxIxzLs δ(r). (2) we have studied the metadimers arranged in an infinite dz two-dimensional periodic lattice. The electromagnetic coupling between the adjacent metadimers was found not Comparing Eq. (1) with Eq. (2), we note that in the ←→ to hinder the realization of the electric dipole suppres- J excitation the non-zero multipole moments are q = xz sion, but simply to red-shift its spectral location. For (ˆxˆz + ˆzˆx)iI Ls/(2ω) and m = ˆyI Ls/2. Another con- xz xz example, for a square lattice with a period of 50 nm, tribution to the electric current excitation present in the ←→ we find the electric dipole suppression to take place at metadimer is Jzx, in which we have a similar q = λ0 = 618 nm. The electric current distribution (includ- (ˆxˆz + ˆzˆx)iI Ls/(2ω), but opposite m = −ˆyI Ls/2, zx zx ing both the conduction and polarization currents in the where, however, I is small compared to I . zx xz silver discs and in the surrounding dielectric) within a sin- It is of interest to compare the field created by the cur- gle unit cell of the lattice is depicted in Fig. 5. The plot rent quadrupole excitation in the metadimer to the field corresponds to the instant of time at which the electric created by an electric dipole. In spherical coordinates, currents have maximum values. The excitation in the the electric far-field of a point electric dipole oriented metadimer exhibits zero electric dipole moment and is along the x-axis and situated at the origin of the coordi- characterized by electric quadrupole and magnetic dipole nate system is moments of considerable strengths. In the figure, we also plot the amplitude of the magnetic field normalized to 2 ikr p k px e that of the incident wave. Within the metadimer, the E (r)= θˆcos θ cos φ − φˆsin φ. (3) F 4πǫ r magnetic field is seen to be significantly enhanced due to the strong opposite electric currents. These calculations On the other hand, the electric far-field of the point cur- clearly indicate the possibility to obtain a metamaterial rent quadrupole introduced in Eq. (2) can be calculated with dominating electric quadrupole and magnetic dipole 4 excitations. for the metamaterial layer to be treated as optically homogeneous. We anticipate that practical realizations of the proposed metamaterial will not only help verify theoretical predictions of light interaction with higher-order multipole materials [19, 20], but also lead to discovering new optical phenomena. Furthermore, we believe that the effects related to higher-order multipoles, such as optical activity, gyrotropic birefringence and dynamic magneto- electric effect [1], can be substantially enhanced and con- trolled in quadrupole metamaterials designed specifically for these purposes. This work was funded by the Academy of Finland (project 134029).

FIG. 5. The electric current distribution and the magnetic field amplitude in an individual silver metadimer belonging [1] R. E. Raab and O. L. de Lange, Multipole Theory in to a two-dimensional square lattice of dimers embedded in Electromagnetism (Oxford, New York, 2005). glass with a period Λ = 50 nm. The illumination wavelength [2] V. M. Shalaev, Nat. Photonics 1, 41 (2007). is λ0 = 618 nm. Arrows: electric current density vectors. Col- [3] D. J. Cho, F. Wang, X. Zhang, and Y. R. Shen, ors: absolute value of the complex magnetic field normalized Phys. Rev. B 78, 121101 (2008). to that of the incident wave. [4] E. B. Graham and R. E. Raab, Proc. R. Soc. Lond. A 456, 1193 (2000). [5] E. B. Graham and R. E. Raab, Proc. R. Soc. Lond. A As is evident from the symmetry of the metadimers, 457, 471 (2001). the response of the metamaterial layer is insensitive to [6] A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, the polarization of light at normal incidence. Another A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, Nature interesting property of the metadimer lattice is that the 438, 335 (2005). reversal of the illumination direction makes the dipole [7] Y. Ekinci, A. Christ, M. Agio, O. J. F. Martin, H. H. 16 suppression phenomenon disappear and the lattice acts Solak, and J. F. Löffler, Opt. Express , 13287 (2008). [8] T. Pakizeh, A. Dmitriev, M. S. Abrishamian, N. Gran- essentially as a lattice of electric dipoles. This can be payeh, and M. Käll, J. Opt. Soc. Am. B 25, 659 (2008). used to create ultrathin bifacial optical components [18]. [9] B. Kante, K. O’Brien, A. Niv, X. Yin, and X. Zhang, A three-dimensional quadrupole metamaterial can be Phys. Rev. B 85, 041103 (2012). obtained by stacking several layers of metadimers on top [10] S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, 101 of each other. Owing to the geometrical simplicity of Phys. Rev. Lett. , 047401 (2008). the proposed metadimers, the metamaterials composed [11] J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and of them should be relatively easy to fabricate. This would F. Capasso, Science 328, 1135 (2010). open up a possibility to study the properties of the still [12] S. I. Bozhevolnyi, A. B. Evlyukhin, A. Pors, unexplored higher-order metamaterials also experimen- M. G. Nielsen, M. Willatzen, and O. Albrektsen, tally. New Journal of Physics 13, 023034 (2011). In summary, we have introduced a concept of [13] C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, quadrupole metamaterials in which electric quadrupoles C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, Phys. Rev. B 83, 245119 (2011). and magnetic dipoles entirely determine the light-matter [14] P. B. Johnson and R. W. Christy, interaction phenomena. We have proposed a realistic Phys. Rev. B 6, 4370 (1972). meta-atom, in the form of a metal-disc nanodimer that [15] J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, can be used to construct such a metamaterial for oper- New York, 1999). ation at certain wavelengths of visible light. The spec- [16] G. Russakoff, Am. J. Phys. 38, 1188 (1970). tral location at which the electric dipole contribution is [17] R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961). suppressed can be changed by tuning, e.g., the disc sep- [18] J. H. Lee, Y.-W. Song, K. H. Hwang, J. gu Lee, J. Ha, aration. Our calculations show that dominating higher- and D.-S. Zang, Opt. Express 16, 16867 (2008). order multipole scattering by the metadimers can also be [19] M. J. Gunning and R. E. Raab, achieved in a periodic two-dimensional array. The size of J. Opt. Soc. Am. B 14, 1692 (1997). the metadimers and the period of the lattice were chosen [20] O. L. de Lange and R. E. Raab, to be much smaller than the wavelength, which allows Am. J. Phys. 74, 301 (2006).