Light scattering by coupled oriented dipoles: Decomposition of the scattering matrix

We study the optical response of two coupled oriented dipoles with the dimer axis perpendicular to the wave vector of light by analyzing how their scattering matrix can be decomposed. The scattering matrix can be written as a linear combination of three terms with a clear physical meaning: one for each particle and another that is responsible for the coupling and that vanishes for noninteracting or distant particles. We show that the interaction term may generate optical activity for certain scattering directions and that this effect manifests itself mostly in the near field. This simple and intuitive theory based on matrix and vector states of oriented dipoles also describes hybridization processes and Fano resonances. The decomposition method can be also formulated in terms of a hybrid basis that allows us to quantitatively determine the individual contribution of the in-phase and out-of-phase coupling modes to the overall intensity. Our method can help to understand the optical response of more complex nanostructures that can be decomposed into dipole terms. The results are illustrated in gold nanoantenna dimers which exhibit a strong dipolar resonance.

Figure 1

Scheme of the geometry. Dimer axis and dipole vectors are perpendicular to the direction of propagation of incident light.

Figure 2

Simulated response of two equal gold nanoantennas forming a cross by ${45}^{\circ }$. (a) shows the geometry and dimensions of the cross structure, which has a thickness of 50 nm. (b) and (c) show the scattered field ${\left|E\right|}^2$ in a plane at 10 nm above the cross when the structure is illuminated with a plane wave of 1640 nm with right-and left-circular polarizations, respectively. In (d) we calculate the ratio $(I_{LCP}-I_{RCP})/(I_{LCP}+I_{RCP})$.

Figure 3

Calculated intensity for the scattering of two coupled dipoles as a function of the distance between them. The dipoles can only polarize along the orientations shown by the arrows drawn at the top of each panel. These calculations correspond to illumination with a left-handed circularly polarized plane wave. The two particles were assumed to have polarizabilities of the same magnitude $({\alpha }_1={\alpha }_2)$ but different orientation. The spectroscopic values of the polarizability that we have used in this example are those that result from applying Clausius Mossotti relation to spherical silver particles in vacuum with a radii of 1 nm and using the Drude model of silver. Note that these spectroscopic values of polarizability are chosen for illustration purposes only and that the calculation is not describing a coupled system of spheres.

Figure 4

Energy splitting in hybrid modes for a dipole dimer perpendicular ($\pi$-type stacking) and parallel ($\sigma$-type stacking) to the dimer axis. Red and blue colors, respectively, indicate positive and negative charge distribution.

Figure 5

Scattering intensity as a function of the distance for two parallel dipoles with different polarizabilities. Because of the interference, there is Fano dip (dashed line) in between the two resonant modes.

Figure 6

(a) Different geometries of nanoantenna dimers considered in the BEM simulations. The only difference between the considered cases is the distance between the center of the nanoantennas ($d$), which is respectively set at 0 nm, 225 nm, 250 nm, 500 nm, 750 nm, and 1500 nm in A, B, C, D, E, and F. (b) Spectroscopic values of $|g_{\mathrm{int}}^2/g_1{g}_2|$ for the six different configurations.

Figure 7

Complex coefficients ${\nu }^{+}$ and ${\nu }^{-}$ corresponding to simulation A. The vertical lines indicate the position of the hybridization resonances.

Figure 8

Comparison of the in-phase and out-phase calculated intensities ($I^{+}$ and $I^{-}$, respectively) with respect to the overall simulated scattering intensity ($I$) for simulations A (a) and B (b).