Giant in-particle field concentration and Fano resonances at light scattering by high-refractive-index particles

We present the results of a detailed analytical study of light scattering by a particle with high refractive index m + i kappa and low losses (m >> 1, 0 < kappa << 1) based on the exact Mie solution. We show that there is a dramatic difference in the behavior of the electromagnetic field within the particle (inner problem) and outside it (outer problem). With an increase in m at fixed values of the other parameters, the field within the particle asymptotically converges to a periodic function of m. The electric and magnetic type Mie resonances of different orders overlap substantially. It may lead to a giant concentration of the electromagnetic energy within the particle. At the same time, we demonstrate that the solution for the outer problem makes it possible to present each partial scattered wave as a sum of two partitions. One of them corresponds to the m-independent wave, scattered by a perfectly reflecting particle and plays the role of a background, while the other is associated with the excitation of a sharply m-dependent resonant Mie mode. The interference of the partitions brings about a typical asymmetric Fano profile. The profile is obtained from the exact Mie solution by means of identical transformations without any additional assumptions and/or fitting. It makes it possible to generalize rigorously the Fano theory to the case of finite dissipation. At an increase in m the Fano resonances in the outer problem die out and the scattered field converges to the universal, m-independent profile. The behavior of the resonances at a fixed m and varying particle size parameter (x) is also discussed in detail. The similarities and differences of the two cases (fixed x, varying m and fixed m, varying x) are disclosed. We also show that under certain very general conditions the scattering cross section of a large lossy sphere cannot be smaller than half its geometric cross section, while its absorption cross section cannot exceed three halves of the geometric one. Numerical estimates of most discussed effects for a gallium phosphide particle irradiated by the second harmonic of a Nd:YAG laser are presented as an example. In addition to purely academic interest, the obtained results may be employed to design new highly nonlinear heterogenic nanostructures and other metamaterials.