Supplementary Information Symmetry Breaking and Optical Negative Index of Closed Nanorings Boubacar Kanté 1, Yong-Shik Park 1, Kevin O Brien 1, Daniel Shuldman 1, Norberto D. Lanzillotti Kimura 1, Zi Jing Wong 1, Xiaobo Yin 1, and Xiang Zhang 1,2 1 NSF Nanoscale Science and Engineering Centre, 3112 Etcheverry Hall, University of California, Berkeley, California 94720, USA 2 Materials Sciences Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA
Supplementary Figure S1 Near fields, remittances, effective indices, and impedance of the coupled nanorings. Simulation results for two layers of coupled nanorings (interlayer 50nm) as well as corresponding mode pattern (E z component of the total electric field at the bottom surface of the top and bottom rings) around telecommunication wavelength. The structures being polarization independent due to symmetry considerations (see Figure S2 to S5), calculations presented here are performed for an electric field parallel to one side of the rings. ω - s=50, aligned, and ω + s=50, aligned correspond to the anti-symmetric and symmetric modes respectively when the rings are aligned, and, ω 3 is a multipole. ω - s=50, shifted, and ω + s=50, shifted correspond to the symmetric and anti-symmetric modes respectively when the rings are fully shifted. (a) and (b): Remittances (Transmission, red, Reflection, black and Absorption blue) of aligned and shifted nanorings. The dotted lines are phases (transmission, red and, reflection, black) in degree. (c) and (d): Near field pattern at transmission dips indicated on the top and
bottom rings. The white arrows indicate the effective electric dipole on each ring, and reveal the symmetric (electric dipole moment) and antisymmetric (magnetic dipole moment) nature of the modes. (e) Effective index when the nanorings are aligned: the index is always positive. (f) Effective index when the nanorings are fully shifted: a negative index appears around the antisymmetric mode. (g) Real (continuous line) and imaginary (dashed lines) parts of the permittivity (blue) and the permeability (black). The inset on the graph (bottom right and corner) shows that the permittivity and permeability are simultaneously negative around 1.5μm. (h) Real and imaginary parts of the normalized impedance. The energy density in the structure can be estimated for a dispersive medium from the relation U = 1 4 d( ωεε ) E dω d( ωµµ 0 + dω 0 2 ) H 2 (see ref. 2 of the paper). Supplementary Figure S2 Polarization resolved measurements for 2 rings layer chess metamaterial. The last layer of rings is covered by SU-8. Angles in degree with respect to one side of the rings.
Supplementary Figure S3 Polarization resolved measurements for 3 rings layer chess metamaterial. The last layer of rings is covered by SU-8. Angles in degree with respect to one side of the rings. Supplementary Figure S4 Polarization resolved measurements for 4 rings layer chess metamaterial. The last layer of rings is covered by SU-8. Angles in degree with respect to one side of the rings.
Supplementary Figure S5 Polarization resolved measurements for 5 rings layer chess metamaterial. The last layer of rings is covered by SU-8. Angles in degree with respect to one side of the rings. Supplementary Figure S6 Side view of the two layer sample. Schematic (plane perpendicular to the rings plane) of the two layer sample measured in Fig. 2.
Supplementary Figure S7 Polarization resolved measurements for 2rings layer without SU-8 on top of the last layer of rings. This corresponds to the data of the sample measured on Fig. 2 of the paper. The antisymmetric mode occurs around 1.9μm. Angles are in degree with respect to one side of the rings.
Supplementary Figure S8 Broadband phase measurement of the multilayer (4 rings layer) with a white light Mach Zehnder interferometer coupled to a spectrometer. The cut-off of the spectrometer is at 1.7μm. Experimental normal incidence transmission (a), transmission phase (c) and index (e). Simulated normal incidence transmission (b), transmission phase (d) and effective index (f). The arrows indicate the zero crossing wavelengths for the phase (index). Repeated measurements gave an error in phase smaller than 0.1 radians, corresponding to an index variation smaller than ± 0.08. Error bars are added to Fig. S10 (e). The transmission phase crosses zero at the dip of the transmission where the index transition from positive to negative values. The effective index around the transition point is simply defined as n eff =-Φ/k 0 d. Good qualitative agreement is found between experiment and theory and in particular, the phase (index) crosses zero at the dip of the transmission. Negative index has thus been directly observed from experiment in a multilayer metamaterial. To lift phase ambiguities, the interferogram can be measured around a wavelength where the index is known to be close to zero (from numerical simulations) with in phase interferograms and the wavelength is varied so as to observe the continuous variation of the delays. Comparison of the trends from the anchor point (zero phase) with simulations allows us to determine whether we have a phase lead or lag.
Supplementary Figure S9 Transmission spectra for 10, 20, 30, and 40 layers of the rings. The broadband negative index band for rings 5 layers in the paper is already in coincidence with higher number of layers. A Drude model is used in simulations with the following parameters: ω p =1.37e16s -1 (plasma frequency) and ω c =8.143e13s -1 (collision frequency). Increasing the number of layer leads to the formation of the band as shown on Figure S11.
Supplementary Figure S10 Transmission spectrum of 20 layers of rings resonators in the aligned configuration (red curve) and with the broken symmetry introduced in the paper (black curve). In the first case, a bandgap is formed in the three dimensional system. In the second case, an ultra-broad Fano induced passband is observed between 1.3μm and 2.3μm with negative index (see paper). This figure demonstrates the fundamental role of symmetry breaking in inducing the negative index band. Supplementary Figure S11 Band formation when the number of layer is increased. The fully symmetric mode is always the lower energy mode, and, the fully antisymmetric mode is the higher energy mode (for the shifted rings). Partially symmetric and antisymmetric modes have intermediate energy levels. Energy increases from bottom to top.
Supplementary Figure S12 Dispersion and figure of merit. Effective parameters (real and imaginary parts) for 10 layers of rings (a) as well as corresponding figure of merit (b) demonstrating the low loss and broadband behavior of our structure. In thin metamaterials, the localized modes reradiate in free space and the radiation loss is high. In thicker samples, the loss is decreased because the radiation from different rings cancelled inside the structure when they are out of phase and leads to decreased radiation loss of the bulk metamaterial. For a fair comparison with the fishnet structure in term of negative index band, we calculated the effective index for 10 metallic layers as reported in ref. 14 of the paper (Valentine et al.). While the index was negative over a band of about 300nm in ref. 14, the negative index of the chess metamaterial spans over about an octave on a band of about 1000nm around 1.5μm resulting in an ultra-broad negative index band with low loss. The low overall transmission is due to the impedance mismatch between the incident medium (air) and the metamaterial.
Supplementary Figure S13 Impedance and remittances of the bulk chess metamaterial. a, Normalized impedance for 20 layers of rings presented in Fig. 5c as well as, b, the corresponding transmission and reflection coefficients. The high reflection coefficient (small transmission coefficient) is due to impedance mismatch. Supplementary Figure S14 Dispersion of shifted and aligned rings. a, Calculated dispersion curves of an infinite array of chess metamaterials around the frequency of interest. The negative index band can be observed. b, Dispersion curves of an infinite array of aligned closed rings, no negative index band is observed. In the two cases, the metallic structures are embedded in a SU-8 background. On both graphs, the horizontal axis is the phase advance per unit cell or kd, where k is the wavevector, and, d is the thickness of the unit cell.