  We analyze the inversion of the photo-acoustic imaging modality using
electromagnetic plasmonic nano-particles as contrast agents. We prove that the
generated pressure, before and after injecting the plasmonic nano-particles,
measured at a single point, located away from the targeted inhomogeneity to
image, and at a given band of incident frequencies is enough to reconstruct the
(eventually complex valued) permittivity. Indeed, from these measurements, we
define an indicator function which depends on the used incident frequency and
the time of measurement. This indicator function has differentiating behaviors
in terms of both time and frequency. First, from the behavior in terms of time,
we can estimate the arrival time of the pressure from which we can localize the
injected nano-particle. Second, we show that this indicator function has
maximum picks at incident frequencies close to the plasmonic resonances. This
allows us to estimate these resonances from which we construct the
permittivity.
  To justify these properties, we derive the dominant electric field created by
the injected nano-particle when the incident frequency is close to plasmonic
resonances. This is done for the full Maxwell system. To this end, we use a
natural spectral decomposition of the vector space (L^2(D))^3 based on the
spectra of the Newtonian and the Magnetization operators. As another key
argument, we provide the singularity analysis of the Green's tensor of the
Maxwell problem with varying permittivity. Such singularity is unusual if
compared to the ones of the elliptic case (as the acoustic or elastic models).
In addition, we show how the derived approximation of the electric fields
propagates, as a source, in the induced pressure with the time.
