  We propose a design concept for tailoring the local density of optical states
(LDOS) in dielectric nanostructures, based on the phase distribution of the
scattered optical fields induced by point-like emitters. First we demonstrate
that the LDOS can be expressed in terms of a coherent summation of constructive
and destructive contributions. By using an iterative approach, dielectric
nanostructures can be designed to effectively remove the destructive terms. In
this way dielectric Mie resonators, featuring low LDOS for electric dipoles,
can be reshaped to enable enhancements of three orders of magnitude. To
demonstrate the generality of the method, we also design nanocavities that
enhance the radiated power of a circular dipole, a quadrupole and an arbitrary
collection of coherent dipoles. Our concept provides a powerful tool for
high-performance dielectric resonators, and affords fundamental insights into
light-matter coupling at the nanoscale.
