Magnetic Mode Coupling in Hyperbolic Bowtie Meta-Antennas

Hyperbolic metaparticles have emerged as the next step in metamaterial applications, providing tunable electromagnetic properties on demand. However, coupling of optical modes in hyperbolic meta-antennas has not been explored. Here, we present in detail the magnetic and electric dipolar modes supported by a hyperbolic bowtie meta-antenna and clearly demonstrate the existence of two magnetic coupling regimes in such hyperbolic systems. The coupling nature is shown to depend on the interplay of the magnetic dipole moments, controlled by the meta-antenna effective permittivity and nanogap size. In parallel, the meta-antenna effective permittivity offers fine control over the electrical field spatial distribution. Our work highlights new coupling mechanisms between hyperbolic systems that have not been reported before, with a detailed study of the magnetic coupling nature, as a function of the structural parameters of the hyperbolic meta-antenna, which opens the route toward a range of applications from magnetic nanolight sources to chiral quantum optics and quantum interfaces.

Using effective medium approach, we calculated the optical response of the composite medium made of alternating layers of metal (Au) and dielectric (TiO2) (Figure S1a).Bruggeman's effective medium theory considers mixture of two materials and can be applied for the composites with arbitrary volume fractions.According to the literature, the hyperbolic metamaterials serve as a uniaxial medium with permittivity given by a tensor ε = [εxx, εyy, εzz], where the in-plane components are defined as εxx = εyy = ε∥, and the out-of-the-plane component describes by εzz= ε⊥.Then, we were looking for the effective permittivity tensor of the Au/TiO2 multilayer system depending on Au fill factor: where   and   are the total thickness of metal and dielectric layers, respectively.Given generalized Maxwell's Equations and using electromagnetic field boundary conditions, the parallel and perpendicular components of the effective permittivity in a multilayer system can be written as follows: where εAu and εd describe the relative permittivity of the metal and dielectric layers, respectively.As represented in Figure S1(b), the effective permittivity of the Au/TiO2 multilayer structure of  Au = 0.83 is different from that of pure gold, revealing the hyperbolic behavior in the region >780 nm.

Figure S1 .
Figure S1.(a) Schematic illustration of hyperbolic multilayer system made of ten alternating layers of Au and TiO2.(b) The real and imaginary parts of the effective permittivity for the Au/TiO2 multilayer nanostructure ( Au = 0.83) determined by effective media theory (EMT).

Figure S2 .
Figure S2.(a) Calculated optical cross sections of Au bowtie nanoantenna with the total thickness of 120 nm under x-polarization illumination.Surface charge distribution in x-z plane (b), field enhancement in x-y plane (c), and current densities superimposed on the electric field enhancement in x-z plane (d) for the mode (iii) at 764 nm.The x-y and x-z planes have been plotted at middle height and middle width for each bowtie.

Figure S3 .
Figure S3.(a) Calculated optical cross sections of metaparticle (triangular nanoprism, f_Au=0.83)under different polarizations.(b) The colormap of extinction cross sections as a function gold fill factor for the hyperbolic metaparticle under x-polarization illumination.(c) Magnetic field enhancement in y-z plane for the magnetic mode under y-polarization.(d) Magnetic field enhancement in x-z plane for the magnetic mode under x-polarization.

Figure S4 .
Figure S4.(a, b) Electric near-field enhancement of the hyperbolic multilayer Au/TiO2 bowtie meta-antenna (fAu=0.83)at magnetic mode wavelength (λⅱ = 1437 nm).The x-y and x-z planes have been plotted at middle height and middle width for each bowtie.

Figure S5 .
Figure S5.(a) Absorption (a) and scattering (b) intensities and extinction (c) cross-section as a function of structure thickness for a metal fill factor of 0.83.The colormap shows the extinction cross-section as a function of metaantenna height using EMT.The data points show the positions of the corresponding resonances for the full multilayer bowtie meta-antennas.

Figure S6 .Figure S7 .
Figure S6.(a) Normalized electric field scan along x axis averaged over the height of meta-antenna for the different fill factors of the metamaterial.Inset: ratio of the electric field inside the elements of the meta-antenna to the total field in scan.(b) Absorption of the electric dipolar mode for the meta-antennas of different fill factors (extracted from Fig. 2c)

Figure S9 .
Figure S9.Magnetic field enhancement in x-z plane for the bow-tie meta-antenna with  Au = 0.83 (a,b) and  Au = 0.33 (c,d) under x-polarized (a,c) and y-polarized (b,d) excitation.

Figure S10 .
Figure S10.The current densities superimposed on magnetic field-amplitude distributions of the magnetic modes for (a) bowtie meta-antennas and (b) individual metaparticles with the Au fill factor ( Au ) of 0.83 at different refractive indices of the dielectric layer from  = 1 to  = 1.5, 2.2, 2.5, 3 and 4.