Singular analysis of Fano resonances in plasmonic nanostructures

The scattering properties of plasmonic nanostructures arranged in two-dimensional periodic arrays are analyzed by expanding their response into perfectly emitting and absorbing modes. It is shown that the frequencies of these modes determine the shape of the reflection and transmission spectra in the same way that the positions of point-like charges determine the electric field around them. This helps to develop a visual interpretation of many resonant effects which occur due to overlapping resonances, and the Fano interference effect is considered as a basic example. This approach works naturally in the domain of complex frequencies and provides a systematic tool to analyze, optimize, and engineer the resonant properties of nanostructures for various applications.

Figure 1

Schematic of a 2D grating composed of dolmen-like structures. The boundaries of elementary cells are highlighted in white, and the semitransparent box around the cell in the middle shows the abstract scattering channels. The periods of the grating are $a_x=a_y=500$ nm. Each cell contains three bars of gold with dimensions $d_1=300$ nm $\times$ $d_2=80$ nm and thickness $d_z=40$ nm. The gaps between bars in the same cell are $g_x=160$ nm and $g_y=30$ nm. The structure is surrounded by air, and the permittivity of the bars is described by the Drude model $\varepsilon \left(\omega \right)=1-{\omega }_p^2/({\omega }^2+i\Gamma \omega )$ with the parameters $\hslash {\omega }_p=9.0$ eV and $\Gamma =0.009{\omega }_p$ [7, 17].

Figure 2

Reflection spectrum of the structure in Fig. 1. The normal incidence is considered, with the electric field polarized along the $x$ axis. The results of the finite-element method are plotted as filled circles, while the results of the analytical formulas, (8) and (10), are shown as the solid line. Dashed lines indicate the positions of the emitting modes ${\omega }_1^{-}$ and ${\omega }_2^{-}$. Their contributions to the spectrum are shown by contours in the background. Insets (top): Mode profiles in the plane $z=0$.

Figure 3

$S_{\mathrm{e}}\left(\omega \right)$ component as a function of complex frequency for the structure in Fig. 1. The positions of poles ${\omega }_r^{-}$ and zeros ${\omega }_r^{+}$ are labeled. Contours of blue and green shades show the absolute value of $S_{\mathrm{e}}$ on a logarithmic scale. White contour lines give the phase of $S_{\mathrm{e}}$ in steps of $\pi$ for thick lines and $\pi /10$ for thin lines.

Figure 4

Contour map of the scattering term $S=(\xi -i)/(\xi +i)$ created by an isolated pair of pole and zero. The use of normalized units assures that every two contours of constant phase $\arg \left(S\right)=C$ and $\arg \left(S\right)=C+\pi$ form a circle which always passes through the pole and zero. The positions of the dip and peak in the reflection spectrum ${\left|R\right|}^2={|(S{\mathrm{e}}^{\mathrm{i}\Delta \phi }-1)/2|}^2$ can be predicted as the intersection points of the circle $C=-\Delta \phi$ with the real axis. One such circle, for $\Delta \phi =0.3\pi$, is shown with a greater thickness. When the phase mismatch between even and odd modes disappears, $\Delta \phi =0$, the spectrum acquires a Lorentzian shape.