Extreme coupling: A route towards local magnetic metamaterials

Genuinely homogeneous metamaterials, which may be characterized by local effective constitutive relations, are required for many spectacular metamaterial applications. Such metamaterials have to be made of meta-atoms, i.e., subwavelength resonators, which exhibit only electric and or magnetic dipole and negligible higher-order multipolar polarizabilities in the spectral range of interest. Here, we show that these desired meta-atoms can be designed by exploiting the extreme coupling regime. Appropriate meta-atoms are identified by performing a multipole analysis of the field scattered from the respective meta-atom. To design those particular meta-atoms it is important to disclose the frequency and angular-dependent polarizability of both dipole moments. We demonstrate the applicability of a purely analytical model to accurately calculate for a normally incident plane wave reflection and transmission from meta-surfaces made of periodically arranged meta-atoms. With our work we identify a possible route towards the engineering of artificial materials while only considering the response from its constituents. Our approach is generally applicable to all spectral domains and can be used to evaluate and design metamaterials made from different constituting materials, e.g., metals, dielectrics, or semiconductors.

Figure 1

(a) Geometry of the cut-plate pair (CPP). The CPP can be understood as a finite metal-insulator-metal waveguide resonator as indicated by the semitransparent film. The diameter of the CPP is fixed to 180 nm. The metal layers are made of gold [51]. The refractive index of the spacer film is assumed to be $1.5$. (b), (c) Imaginary parts of the induced electric and magnetic dipole moments $p_{\mathrm{x}}$ and $m_{\mathrm{z}}$ for different gap thicknesses $d$ and fixed metal-layer thickness $d_{\mathrm{Au}}=30$ nm. The legend indicates the thickness $d$. (d), (e) Imaginary parts of the induced electric and magnetic dipole moments $p_{\mathrm{x}}$ and $m_{\mathrm{z}}$ for different metal-layer thickness $d_{\mathrm{Au}}$ and fixed gap thickness $d=5$ nm. The legend indicates the thickness $d_{\mathrm{Au}}$ in nm. Each of the curves is normalized to the results obtained for a CPP with $d=5$ nm and $d_{\mathrm{Au}}=30$ nm. The CPPs are illuminated with an $x$-polarized plane wave propagating in the $y$ direction. For the ratio $\lambda /D$ the free space wavelength is used. Note the different frequency axes in (b), (c) and (d), (e) for better resolution of the resonance shift.

Figure 2

(a) Logarithmic plot of the scattering cross section normalized to ${\omega }^4$. The solid curves indicate the electric (blue) and magnetic dipole (red) as well as the electric (green) and the magnetic quadrupole (cyan) contributions for the CPP. The dotted curves trace the values of each multipolar moment at the magnetic resonance while changing the gap thickness $d$ from 30 nm to 1 nm. (b), (c) Real and imaginary part of the tangential electric and magnetic dipole moment $p_{\mathrm{t}}=(p_{\mathrm{x}},p_{\mathrm{z}})$ and $m_{\mathrm{t}}=(m_{\mathrm{x}},m_{\mathrm{z}})$ when changing the angle of incidence ($0^{\circ },{45}^{\circ },{90}^{\circ }$ with respect to the axis of rotational symmetry) or the polarization (TE and TM polarization is used). The CPP used here is characterized by $D=180$ nm, $d=5$ nm and $d_{\mathrm{Au}}=30$ nm. The different lines obtained for different polarizations and angles of incidence coincide very well for small frequencies, where the deviation gets larger with increasing frequency. Note the normalization of the dipole moments to the exciting field.

Figure 3

The modulus of the total electric field inside the CPP at the magnetic-mode resonance for different distances between the plates: (a) 30 nm, (b) 5 nm, and (c) 1 nm.

Figure 4

(a) Logarithmic plot of the scattering cross section normalized to ${\omega }^4$. The solid curves indicate the electric (blue) and magnetic dipole (red) as well as the electric (green) and the magnetic quadrupole (cyan) contributions for the SRR. The dotted curves trace the values of each multipolar moment at the magnetic resonance while changing the split size $g$. (b) Schematic drawing of the split-ring resonator. The thickness is fixed to $d=30$ nm. The width of the quadratic SRR is fixed to $W=180$ nm. The split gap size $g$ is varied. (c) Modulus of the magnetic dipole moment for changing split gap sizes as indicated in the legend. For illumination a $y$-polarized plane wave propagating in the $z$ direction is used.

Figure 5

Reflectance (black) and transmittance (blue) at a periodic ensemble of cut-plate pairs on a square lattice. The solid lines are the results obtained from rigorous simulations. The dashed lines are for the analytical model based on the dipole polarizabilities. The cut-plate plate has a diameter of $D=180\mathrm{nm}$, a metal layer thickness of $d_{\mathrm{Au}}=30\mathrm{nm}$ and a spacer thickness $d$ of $5\mathrm{nm}$ (a) and $30\mathrm{nm}$ (b) respectively.