Intrinsic multipolar contents of nanoresonators for tailored scattering

We introduce a theoretical and computational method to design resonant objects, such as nanoantennas or meta-atoms, exhibiting tailored multipolar responses. In contrast with common approaches that rely on a multipolar analysis of the scattering response of an object upon specific excitations, we propose to engineer the intrinsic (i.e., excitation-independent) multipolar content and spectral characteristics of the natural resonances—or quasinormal modes—of the object. A rigorous numerical approach for the multipolar decomposition of resonances at complex frequencies is presented, along with an analytical model conveying a direct physical insight into the multipole moments induced in the resonator. Our design strategy is illustrated by designing a subwavelength optical resonator exhibiting a Janus resonance that provides side-dependent coupling to waveguides over the full linewidth of the resonance and on a wide angular range for linearly polarized incident plane waves. The method applies to all kinds of waves and may open new perspectives for subwavelength-scale manipulation of scattering and emission.

Figure 1

Intrinsic multipolar content of resonances. The plasmonic dolmen is made of silver and composed of an upper rod ($128s\times 50s\times 20{\mathrm{nm}}^3$) separated by a gap of width $g$ from two lower rods ($30\times 100\times 20{\mathrm{nm}}^3$) separated by 30 nm. The silver permittivity is approximated by a single-pole Drude-Lorentz model with ${\epsilon }_{\infty }=1$, ${\omega }_p=1.366\times {10}^{16}{\mathrm{rad}\mathrm{s}}^{-1}$, and $\gamma =0.0023{\omega }_p$. The dolmen is placed in air ($n_{\text{b}}=\sqrt{{\epsilon }_{\text{b}}}=1$). (a) Complex-frequency plane of the plasmonic dolmen with $g=30$ nm and $s=1$. (b) Spatial maps of $|{\tilde{\mathbf{E}}}_j|$ of the three QNMs found in the visible range ($j=1$ to 3). Each QNM behaves as a superposition of electric and magnetic multipoles, as suggested by the current densities (white arrows). The high-field enhancement driven by plasmonic effects in the near-field region hides the field divergence due to the complex frequency. (c) Induced dipole moments $\text{Re}\left[p_x\right]$ and $-\text{Re}\left[m_z\right]n_{\text{b}}/c$ at real frequencies for a plane-wave excitation ${\mathbf{E}}^{\text{b}}\left(z\right)=E_0\widehat{\mathbf{x}}exp\left[i\omega n_{\text{b}}z/c\right]$. The response is dominated by modes I and II (solid and dashed lines, respectively). Modes III and VII are not excited for symmetry reasons. As shown by a comparison with exact real-frequency calculations (circles), the induced moments are well predicted with the eight QNMs represented in (a) (dotted lines).

Figure 2

Engineering the spectrum and the multipolar content of the plasmonic dolmen resonances. (a) Trajectories of the two dominant QNMs in the complex frequency plane throughout the design, upon tuning the gap width $g$ from 45 to 10 nm (solid arrows) and the size factor $s$ from 1 to 0.4 (dashed arrows). (b) Mode I is mostly composed of an electric dipole and a magnetic dipole. For $g=10$ nm and $s=0.5125$, one obtains $|{\tilde{p}}_{x,1}|=|{\tilde{m}}_{z,1}|n_{\text{b}}/c$ and a phase difference $\mathrm{\Delta }\phi \approx -\pi /2$ (not shown). (c) Induced dipole moments at real frequencies for an $x$-polarized plane-wave excitation, retrieved from VSWF expansion at real frequencies ($p_x$ in solid lines, $m_z$ in dashed lines). One finds $p_x\approx -i{m}_z{n}_{\text{b}}/c$. The offset (shaded area) in $\text{Re}\left[p_x\right]$ is due to the remnants of mode II and other modes outside the visible range; see the SM [35].

Figure 3

Demonstration of Janus effect in scattering configuration. (a) Sketch of the studied configuration. The dolmen is placed at 117 nm (border to border) above a dielectric nanowire of index 2.03 with dimensions $270\times 200{\mathrm{nm}}^2$ along $y$ and $z$ and invariant along $x$, and is illuminated by an $x$-polarized plane wave for varying wavelengths and incident angles ${\theta }_{\text{i}}$. (b) Spatial maps for the scattered magnetic-field $z$ component at ${\theta }_{\text{i}}=0^{\circ }$ and $\lambda =0.713\mu \mathrm{m}$. Depending on the dolmen orientation, light is either efficiently coupled or uncoupled to the fundamental nanowire mode. (c),(d) Spectral and angular dependence of the coupling efficiencies, evidencing that the Janus effect is effectively implemented for many incidences and frequencies of the driving field.