Reexamination of Kerker's conditions by means of the phase diagram

For passive electromagnetic scatterers, we explore a variety of extreme limits on directional scattering patterns in a phase diagram, regardless of the details on the geometric configurations and material properties. By demonstrating the extinction cross sections with the power conservation intrinsically embedded in the phase diagram, we give an alternative interpretation for Kerker's first and second conditions, associated with zero backward scattering (ZBS) and nearly zero forward scattering (NZFS). Physical boundaries and limitations for these directional radiations are illustrated along with a generalized Kerker's condition with implicit parameters. By taking the dispersion relations of gold silicon core-shell nanoparticles into account, we reveal the realistic parameters to experimentally implement ZBS and NZFS at optical frequencies by means of a phase diagram.

Figure 1

(a) Phase diagram for the extinction cross section, defined by an amplitude square ($|a_n{|}^2,{\left|b_n\right|}^2$) and a phase $[\mathrm{Arg}\left(a_n\right),\mathrm{Arg}\left(b_n\right)]$ of scattering coefficients for passive scatterers. The colored region indicates the allowed solutions with the contour line representing a constant normalized extinction cross-sectional $2\pi {\sigma }_n^{\mathrm{ext}(a,b)}/(2n+1){\lambda }^2$ in transverse magnetic (TM) or transverse electric (TE) modes. Three sets of supported scattering states are marked as (I), (II), and (III) with the corresponding three-dimensional–two-dimensional (3D-2D) radiation patterns depicted in (b)–(g), respectively. Here, (b) and (c) reveal the ZBS condition; (d) and (e) reveal the nearly ZFS condition as a residue of scattering contributions can still be found in the enlarged window in the forward direction (see the inset); (f) and (g) reveal a state with the same extinction cross section as that of (d) and (e) (note the scale) but with losses. The size of the scatterer used in all simulations is the same by setting $a={10}^{-2}\lambda$.

Figure 2

(a) An unperturbed NZFS state marked as (I) in the phase diagram is studied with the introduction of a small perturbation of $\delta \beta$ as marker (II), $\delta \alpha$ as marker (III), and $-\delta \alpha$ as marker (IV), respectively. (b) The corresponding auxiliary complex space, defined by the real and imaginary parts of the scattering coefficient $a_1+b_1$ for the unperturbed and perturbed states. (c) The resulting 2D scattering patterns for the unperturbed and perturbed states given in (a) with the enlarged window in the forward direction shown in (d).

Figure 3

The corresponding spectra for the scattering cross section (the dashed-blue curve), the forward scattering cross section (the solid-red curve), the backward scattering cross section (the dashed-dotted-orange curve), and the absolute values of the two lowest-order scattering coefficients ($|a_1|$ and $|b_1|$ in the green and purple colors, respectively) for a core-shell nanoparticle shown in the inset. This core-shell particle consists of gold in the core and silicon in the shell with the corresponding dispersion relation taken from the experimental data [28].

Figure 4

Locations of the two lowest-order scattering coefficients $a_1$ and $b_1$ at different wavelengths in the phase diagram, generated from the data points depicted in Fig. 3. Here, the green and purple dots represent the scattering coefficients for electric and magnetic dipoles $a_1$ and $b_1$, respectively. The markers ${\lambda }_A$ and ${\lambda }_B$ highlight the two crossing points marked in Fig. 3, i.e., for the ZBS and NZFS conditions, respectively. The corresponding 3D-2D scattering patterns are plotted in (b) and (c) for the ZBS condition (marker ${\lambda }_A$) and in (d) and (e) for the NFZS condition (marker ${\lambda }_B$).