Supporting information for: Hybridization of Lattice Resonances

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1 Supporting information for: Hybridization of Lattice Resonances Sebastian Baur, Stephen Sanders, and Alejandro Manjavacas Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 873, United States S

2 a 3 ii a 3 ij - <,2> <2,> <,2> <2,> <, 2> <2, > <,> <, > Real part Imaginary part = =a/ <,> <,> a 3 ii a 3 ij - <,2> <2,> <,2> <2,> <, 2> <2, > <,> <, > = =a/ = =a/ <,> <,> (d) (e) - - a 3 ij = = (c) a 3 ij =a/ =a/ (f) / /2 5 -/2 5 λ/a -/2 λ/a Figure S: Examples of the spectral dependence of the lattice sum G ij that are relevant to some of the arrays studied in the main paper. In all cases, we consider a square array with periodicity a, and normal incidence. Blue and red curves are used to indicate the real and imaginary parts of the lattice sum, respectively. component of G ii, which is identical to the component. component of G ij for x i =, y i = a/. (c) component of G ij for x i =, y i =. (d) component of G ij for x i =, y i = a/. (e) component of G ij for x i =, y i = a/. (f) component of G ij for x i = a/, y i = a/. S2 -

<,> <,> <,> <, > ( ( ( 3/ /2 / 3/ /2 / 3/ /2 / Re{ ij } - - ( ( ( 3/ /2 / 3/ /2 / 3/ /2 / / /2 3/ (

calculated as a function of the position of particle j, measured with respect to particle i, for

Panel displays the results for the,,, Rayleigh anomaly occurring at λ = a, while displays the

3 <,> <,> <,> <, > ( ( ( 3/ /2 / 3/ /2 / 3/ /2 / Re{ ij } - - ( ( ( 3/ /2 / 3/ /2 / 3/ /2 / / /2 3/ ( - / /2 3/ ( Figure S2: Real part of the lattice sum G ij normalized to the real part of G ii, calculated as a function of the position of particle j, measured with respect to particle i, for normal incidence. Panel displays the results for the,,, Rayleigh anomaly occurring at λ = a, while displays the results for the,,, Rayleigh anomaly, which appears in the spectrum at λ = a/ 2. In both cases, the upper, middle, and lower plots correspond, respectively, to the,, and components of the lattice sum. S3

4 <,> <,> <,> <, > ( ( ( 3/ /2 / 3/ /2 / 3/ /2 / Re{ ij } - - ( ( ( 3/ /2 / 3/ /2 / 3/ /2 / / /2 3/ ( - / /2 3/ ( Figure S3: Same as Figure S2, but calculated at wavelengths λ =.a and λ =.a/ 2. S

<,2> <2,> <2,2> <2, 2> ( ( ( 3/ /2 / 3/ /2 / 3/ /2 / Re{ ij } - - ( ( ( 3/ /2

sum G ij normalized to the real part of G ii, calculated as a function of the

while displays the results for the 2, 2, 2, 2 Rayleigh anomaly, which appears

5 <,2> <2,> <2,2> <2, 2> ( ( ( 3/ /2 / 3/ /2 / 3/ /2 / Re{ ij } - - ( ( ( 3/ /2 / 3/ /2 / 3/ /2 / / /2 3/ ( - / /2 3/ ( Figure S: Real part of the lattice sum G ij normalized to the real part of G ii, calculated as a function of the position of particle j, measured with respect to particle i, for normal incidence. Panel displays the results for the, 2, 2, Rayleigh anomaly occurring at λ =, while displays the results for the 2, 2, 2, 2 Rayleigh anomaly, which appears in the spectrum at λ = a/ 8. In both cases, the upper, middle, and lower plots correspond, respectively, to the,, and components of the lattice sum. S5

<,2> <2,> <, 2> <2, > <3,> <,3> <3, > <, 3> <,5> <5,> (

3/ /2 / 3/ /2 / / /2 3/ ( - / /2 3/ ( Figure S5: Real

of G ii, calculated as a function of the position of

Rayleigh anomaly occurring at λ = a/ 5, while displays

anomaly, which appears in the spectrum at λ = a/5.

6 <,2> <2,> <, 2> <2, > <3,> <,3> <3, > <, 3> <,5> <5,> ( ( ( 3/ /2 / 3/ /2 / 3/ /2 / Re{ ij } - - ( ( ( 3/ /2 / 3/ /2 / 3/ /2 / / /2 3/ ( - / /2 3/ ( Figure S5: Real part of the lattice sum G ij normalized to the real part of G ii, calculated as a function of the position of particle j, measured with respect to particle i, for normal incidence. Panel displays the results for the, 2, 2,,, 2, 2, Rayleigh anomaly occurring at λ = a/ 5, while displays the results for the 3,,, 3, 3,,, 3,, 5, 5, Rayleigh anomaly, which appears in the spectrum at λ = a/5. In both cases, the upper, middle, and lower plots correspond to the,, and components of the lattice sum, respectively. S6

.6 Coupled Dipole Model Finite Element Method.5 Reflectance..2 Reflectance..5 (c) (d). 8 8 82 a/ λ (nm) a/.

Finite Element Method (FEM) simulation, performed using the commercial software COMSOL Multiphysics.

7 .6 Coupled Dipole Model Finite Element Method.5 Reflectance..2 Reflectance..5 (c) (d) a/ λ (nm) a/ Induced charge Field enhancement max -max max min (e) (f) λ (nm) Figure S6: Benchmark of the coupled dipole model against full-wave solutions of Maxwell s equations obtained through a Finite Element Method (FEM) simulation, performed using the commercial software COMSOL Multiphysics. (a,b) Reflectance for the arrays investigated in Figures 3 and 3(e) of the main paper, both of which consist of a square lattice with periodicity a = 8 nm and a unit cell containing two silver nanoshells with silica cores and dimensions R i = nm, R o = 5 nm for particle, and R i = 5 nm, R o = 6 nm for particle 2. The positions of the particles are x =, y =, and x 2 =, y 2 = a/ for the array of panel and x =, y =, and x 2 =, y 2 = for that of panel. The black curves represent the results obtained using the coupled dipole model, while the gray curves indicate the outcome of the FEM simulation. Both cases show an excellent agreement between the predictions of the coupled dipole method and the FEM simulations. The main difference is a shift of the peak position, which we attribute to the effect of the higher order modes not accounted for in the dipole approximation. (c,d) Charge density induced in the nanoshells (c), and field enhancement in the plane of the array (d), both calculated using the FEM approach for the two lattice resonances of the array of panel. (e,f) Same as panels (c) and (d) for the lattice resonance of the array of panel. These results confirm the predictions of Figure 3 of the main paper, and therefore support the validity of the coupled dipole model. S7

.9 Coupled Dipole Model Finite Element Method.6 Reflectance.6.3 Reflectance..2 (c). 8 825 85 875 a/ (d) λ (nm) a/.

8 .9 Coupled Dipole Model Finite Element Method.6 Reflectance.6.3 Reflectance..2 (c) a/ (d) λ (nm) a/. Induced charge Field enhancement max -max max min (e) (f) λ (nm) Figure S7: Same as Figure S6 but for nanoshells with increased sizes. Specifically, we use R i = 8 nm, R o = nm for particle, and R i = nm, R o = 2 nm for particle 2. The rest of the parameters are identical to those of Figure S6. The coupled dipole model results are in good agreement with the FEM simulations. As expected, the contribution of higher order modes is increased with respect to the systems of Figure S6, which leads to a larger shift between the coupled dipole model predictions and the FEM results, although this effect is partially masked by the increased linewidth of the resonances. The charge density and field enhancement maps also confirm the larger contribution of higher order modes. S8

Photonic/Plasmonic Structures from Metallic Nanoparticles in a Glass Matrix

Excerpt from the Proceedings of the COMSOL Conference 2008 Hannover Photonic/Plasmonic Structures from Metallic Nanoparticles in a Glass Matrix O.Kiriyenko,1, W.Hergert 1, S.Wackerow 1, M.Beleites 1 and