Modal representation of spatial coherence in dissipative and resonant photonic systems

We provide a self-consistent electromagnetic theory of the link between spatial coherence and optical resonances in three-dimensional open and dissipative photonic systems. The theory that relies on the concept of quasinormal modes with complex frequencies provides an accurate modal expansion of the imaginary part of the Green tensor that correctly treats the effects of radiative leakage, absorption, and dispersion. It represents a powerful tool for calculating and understanding the degree of spatial coherence in complex photonic or plasmonic systems that are governed by a small number of resonances. Comparisons with fully vectorial calculations evidence the high accuracy of the predictions achieved by our semianalytical treatment in the case of coupled photonic-crystal microcavities and plasmonic nanoantennas made of metallic nanorods.

Figure 1

Coupled photonic-crystal microcavities. Two cavities are formed by removing two cylinders in a finite-size photonic crystal. (a, b) Intensity distributions $|{\tilde{E}}_z{|}^2$ of the two dominant QNMs. White circles represent the semiconductor cylinders forming the photonic crystal. We calculate the CDOS for ${\mathbf{r}}^{\prime }$ fixed in the center of the upper cavity; it is normalized by the LDOS in vacuum. (c) Spectral variation of the CDOS for $\mathbf{r}$ fixed in the center of the bottom cavity. The total CDOS results from two modal contributions, ${\rho }_1$ and ${\rho }_2$, given by Eq. (11) and shown by the solid (blue) and dashed (green) curves. The sum [bold solid (red) curve] is in excellent agreement with fully vectorial calculations of the full Green tensor shown by open black circles. (d) Cross section of the CDOS distribution for $\omega d/\left(2\pi c\right)=0.3940$ [frequency where the CDOS in (c) cancels]. Point $\mathbf{r}$ is varied along the $x=0$ axis. For this particular frequency, both modal contributions interfere destructively everywhere.

Figure 2

Single gold nanorod (diameter $D=30$ nm, length $L=100$ nm) embedded in a host medium of refractive index $n=1.5$. (a) Electric-field distribution $|\widetilde{E_z}|$ of the dipole-like QNM (complex frequency $2\pi c/\tilde{\omega }=920+47i$ nm). We calculate the CDOS for ${\mathbf{r}}^{\prime }$ fixed on the axis at a 10-nm distance above the nanorod; it is normalized by the LDOS in a bulk material with $n=1.5$. (b) Longitudinal cross section of the $z$ component of the CDOS at nanorod resonance $\lambda =920$ nm. Point $\mathbf{r}$ is moved along the rod axis. (c) Transversal cross section of the $z$ component of the CDOS at $\lambda =920$ nm. Point $\mathbf{r}$ is now moved perpendicularly to the rod axis at $z=0$. Dashed vertical lines represent the nanorod boundaries. (d) Spectral variation of the CDOS. In (b)–(d), solid and dashed curves are obtained from Eq. (11) for the QNM shown in (a) and the markers are fully vectorial calculations of the full Green tensor.

Figure 3

Three identical gold nanorods (diameter $D=30$ nm, length $L=100$ nm) embedded in a host medium of refractive index $n=1.5$. The rods are aligned and separated by a small gap, $g=40$ nm. (a) Electric-field distributions $|\widetilde{E_z}|$ of the three QNMs that result from the coupling of the three individual dipole-like modes. (b) Spectral variation of the CDOS. Points $\mathbf{r}$ and ${\mathbf{r}}^{\prime }$ are fixed at the positions marked by the white crosses in (a) (on the axis, inside the gaps, at a 10-nm distance from the left and right nanorods) and the CDOS is normalized by the LDOS in a bulk material with $n=1.5$. The three modal contributions to the CDOS are shown by the dashed-dotted black, dashed (green), and solid (blue) curves. The bold solid (red) curve is the sum and the open black circles show the results of fully vectorial calculations.