  The Purcell effect is usually described as a modification of the spontaneous
decay rate in the presence of a resonator. In plasmonics, this effect is
commonly associated with a large local-field enhancement in "hot spots" due to
the excitation of surface plasmons. However, high-index dielectric
nanostructures, which become the basis of all-dielectric nanophotonics, can not
provide high values of the local-field enhancement due to larger radiation
losses. Here, we demonstrate how to achieve a strong Purcell effect in
all-dielectric nanostructures, and show theoretically that the Purcell factor
can be increased by two orders of magnitude in a finite chain of silicon
nanoparticles. Using the eigenmode analysis for an infinite chain, we
demonstrate that the high Purcell factor regime is associated with a Van Hove
singularity. We perform a proof-of-concept experiment for microwave frequencies
and observe the 65-fold enhancement of the Purcell factor in a chain of 10
dielectric particles.
