Generic singularities of scattering coefficients and a paradox of resonant wave scattering

We show that in the exact solution describing the resonant scattering of a plane monochromatic electromagnetic wave by a lossless spherical or cylindrical particle of radius $R$, the scattering coefficients in the multipolar expansion have the generic singularities at vanishing $R$ and certain values of the particle permittivity. At these singularities, the linewidths of the corresponding resonances turn to zero, while the partial cross sections do not have definite limits and may take any value from a certain domain. The value depends on the trajectory along which one approaches the singularities in the plane of the problem parameters. To resolve the singularity it is required to go beyond the commonly used monochromatic approximation, to take into account the finite linewidth of the incident wave, and then to perform the correct sequence of limit transitions, while the straightforward application of the monochromatic approximation may give rise to erroneous results. The effects of finite dissipation are discussed too. Our study may be important for a broad class of resonant wave scattering phenomena associated with high-$Q$ resonances, in particular to the problem of the incident wave interaction with the bound states in the continuum.

Figure 1

Surface $|a_{\ell }(F_{\ell },G_{\ell }){|}^2=\frac{F_{\ell }^2}{F_{\ell }^2+G_{\ell }^2}$ and a family of lines $F_{\ell }=\text{const}$ on it (thin black lines). The closer $F_{\ell }$ to zero, the steeper the maximum of the lines at $G_{\ell }=0$. Eventually, the two wings of the surface intersect each other along the line $F_{\ell }=G_{\ell }=0$. Though various trajectories lying in the surface may intersect this line at different values of $|a_{\ell }{|}^2$, all points of the intersections are projected onto one and the same point $F_{\ell }=G_{\ell }=0$ in the $(F_{\ell },G_{\ell })$ plane. As an example of these trajectories, a set of the level lines $|a_{\ell }{|}^2=\text{const}$ for different values of the constant are shown as thick white lines.

Figure 2

Mutual orientation of the cylinder, coordinate frame, and vectors $\mathbf{k},\mathbf{E}$, and $\mathbf{H}$ of the incident linearly polarized plane wave.

Figure 3

(a) Contour plot of $|a_1{(q,\varepsilon )|}^2$ calculated according to the exact solution, Eq. (4). (b) Its cross sections by the planes $q=0.1$ (blue), $q=0.05$ (dark yellow), and $q=0.025$ (green), i.e., the corresponding line shapes as the functions of $\varepsilon$. Note the similarity with Fig. 1 and the contraction of the linewidths at $q\to 0$.