  Scanning near-field optical imaging (SNOM) using local active probes provides
in general images of the electric part of the photonic local density of states.
However, certain atomic clusters can supply more information by simultaneously
revealing both the magnetic (m-LDOS) and the electric (e-LDOS) local density of
states in the optical range. For example, nanoparticles doped with rare-earth
elements like europium or terbium provide both electric dipolar (ED) and
magnetic dipolar (MD) transitions. In this theoretical article, we develop a
quantum description of active systems (rare earth ions) coupled to a photonic
nanostructure, by solving the optical Bloch equations together with Maxwell's
equations. This allows us to access the population of the emitting energy
levels for all atoms excited by the incident light, degenerated at the
extremity of the tip of a near-field optical microscope. We show that it is
possible to describe the collected light intensity due to ED and MD transitions
in a scanning configuration. By carrying out simulations on different
experimentally interesting systems, we demonstrate that our formalism can be of
great value for the interpretation of experimental configurations including
various external parameters (laser intensity, polarization and wavelength, the
SNOM probe size, the nature of the sample ...).
