  Pure and bright single photon sources have recently been obtained by
inserting solid-state emitters in photonic nanowires or microcavities. The
cavity approach presents the attractive possibility to greatly increase the
source operation frequency. However, it is perceived as technologically
demanding because the emitter resonance must match the cavity resonance. Here
we show that the spectral matching requirement is actually strongly lifted by
the intrinsic coupling of the emitter to its environment. A single photon
source consisting of a single InGaAs quantum dot inserted in a micropillar
cavity is studied. Phonon coupling results in a large Purcell effect even when
the quantum dot is detuned from the cavity resonance. The phonon-assisted
cavity enhanced emission is shown to be a good single-photon source, with a
brightness exceeding $40$ \% for a detuning range covering 15 cavity
linewidths.
