Quasinormal modes expansions for nanoresonators made of absorbing dielectric materials: study of the role of static modes

TL;DR

The paper addresses how to accurately represent light scattering by nanoresonators made of absorbing dielectrics using quasinormal modes. It identifies static modes with zero eigenfrequency as a source of a non resonant background in the modal expansion, yielding a total scattered field E_s = E_s^{nr} + E_s^{r} where E_s^{nr} arises from static-mode excitation. Through lossless and absorbing silicon sphere examples, it shows that including static modes is essential for correct internal field reconstruction and cross sections, with convergence improving as 1/N^3 when static modes are included. The work reconciles two competing QNM formalisms by highlighting the non resonant background provided by static modes and sets a path for accurate modal expansions in more complex geometries, while linking the non resonant term to causality-based arguments. These results imply that static modes are not merely formal constructs but provide a physically significant background that shapes off resonant and interference phenomena such as Fano resonances in absorbing dielectrics.

Abstract

The interaction of light with photonic resonators is determined by the eigenmodes of the system. Modal theories based on quasinormal modes provide a natural tool to calculate and understand light scattering by nanoresonators. We show that, in the case of resonators made of absorbing dielectric materials, eigenmodes with zero eigenfrequency (static modes) play a key role in the modal formalism. The excitation of static modes builds a non-resonant contribution to the modal expansion of the scattered field. This non-resonant term plays a crucial physical role since it largely contributes to the off-resonance signal to which resonances are added in amplitude, possibly leading to interference phenomena and Fano resonances. By considering light scattering by a silicon nanosphere, we quantify the impact of static modes. This study shows that the importance of static modes is not just formal. Modal expansions without static modes reconstruct an incorrect internal field and incorrect extinction and absorption cross-sections. Static modes are of prime importance in an expansion truncated to only a few modes.

Figures (4)

• Figure 1: QNM theory applied to a lossless sphere (refractive index $n = 4$ and radius $R =75$ nm) embedded in a homogeneous medium ($n_b = 1$) and illuminated by a right-handed circularly polarized plane wave. (a) Spectrum of the extinction cross-section $\sigma_e$ normalized by the sphere apparent surface $\pi R^2$. (b) Spectrum of the average intensity enhancement inside the sphere, $(1/V_s)\int_{V_s} |\mathbf{E}|^2/|\mathbf{E}_i|^2 d^3{\bf r}$. (c) Intensity enhancement $|\mathbf{E}|^2/|\mathbf{E}_i|^2$ at $\omega/c = k_0 = 11.36$$\mu$m$^{-1}$ along the sphere radius for $\theta = 36.23^\circ$ and $\phi = 0$. The results of a QNM expansion that includes static modes (red solid line) are compared with those of a QNM expansion without static modes (blue dashed line). Mie theory (black circles) is used as a reference. In (c), $\Delta$ represents the error made with a modal expansion without static modes. The relative error can be as large as 100%. Modal calculations are made with 500 QNMs + 5 static modes.
• Figure 2: Convergence of the QNM expansion for the lossless sphere of Fig. \ref{['fig:lossless1']}. (a) Positions of a few modes in the complex frequency plane. Modes labeled M1, M2 and M3 are responsible for the two peaks in Fig. \ref{['fig:lossless1']}(a). Circles: electric (TM) modes. Crosses: magnetic (TE) modes. Black square: static modes. Modes with $l=1$ (resp. $l=2$) are shown with green (resp. purple) markers. Modes with $l>2$ are calculated but not shown here, see Supplement 1. (b) Relative error on the cross-section as a function of the number of modes. (c) Error on the total electric field integrated over the sphere volume. Results of Mie theory ($\sigma_e^\textrm{Mie}$ and ${\bf E}^\textrm{Mie}$) are taken as a reference. The convergence of a QNM expansion that includes 5 static modes (red solid line) is compared with the convergence of a QNM expansion without static modes (blue dashed line). The error is calculated at $k_0 = 9.971$$\mu$m$^{-1}$. In (b), the black solid curve shows the mean of the relative error over the spectral range considered in Fig. \ref{['fig:lossless1']} and the thin dash-dotted line displays a $1/N^3$ decrease. In (c), the QNM expansion without static modes does not converge towards the exact result provided by Mie theory. The detailed list of the 500 QNMs used in (b) and (c) can be found in Supplement 1.
• Figure 3: QNM theory applied to an absorbing and dispersive sphere (silicon, radius $R =75$ nm) embedded in a homogeneous medium ($n_b = 1$) and illuminated by a right-handed circularly polarized plane wave. (a) Spectrum of the extinction cross-section $\sigma_e$. (b) Spectrum of the absorption cross-section $\sigma_a$. (c) Intensity enhancement $|\mathbf{E}|^2/|\mathbf{E}_i|^2$ at $\omega/c = k_0 = 11.36$$\mu$m$^{-1}$ along the sphere radius for $\theta = 36.23^\circ$ and $\phi = 0$. The results of a QNM expansion that includes static modes (red solid line) are compared with those of a QNM expansion without static modes (blue dashed line). Mie theory (black circles) is used as a reference. In (b) and (c), the cross-sections are normalized by the sphere apparent surface $\pi R^2$. In (c), $\Delta$ represents the error made with a modal expansion without static modes. The relative error can be larger than 100%. Modal calculations are made with 650 QNMs + 5 static modes.
• Figure 4: Convergence of the QNM expansion for the absorbing sphere of Fig. \ref{['fig:lossy1']}. (a) Positions of a few modes in the complex frequency plane. Circles: electric (TM) modes. Crosses: magnetic (TE) modes. Black square: static modes. Modes with $l=1$ (resp. $l=2$) are shown with green (resp. purple) markers. Modes with $l>2$ are calculated but not shown here, see SI. The accumulation point corresponds to the pole of the permittivity. (b) Relative error on the extinction cross-section as a function of the number of modes. (c) Error on the total electric field integrated over the sphere volume. Results of Mie theory ($\sigma_e^\textrm{Mie}$ and ${\bf E}^\textrm{Mie}$) are taken as a reference. The convergence of a QNM expansion that includes 5 static modes (red solid line) is compared with the convergence of a QNM expansion without static modes (blue dashed line). The error is calculated at $k_0 = 10.267$$\mu$m$^{-1}$. In (b), the black solid curve shows the mean of the relative error over the spectral range considered in Fig. \ref{['fig:lossy1']} and the thin dash-dotted line displays a $1/N^3$ decrease. In (b) and (c), the QNM expansion without static modes does not converge towards the exact result provided by Mie theory. The detailed list of the 650 QNMs used in (b) and (c) can be found in Supplement 1.