  The state of the art in electromagnetic Finite Element Particle-in-Cell
(EM-FEMPIC) has advanced significantly in the last few years. These have
included understanding function spaces that must be used to represent sources
and fields consistently, and how currents should be evolved in space and time.
In concert, these achieve satisfaction of Gauss laws. All of these, were
restricted to conditionally stable explicit time stepping. More recently, there
has been advances to the state of art. It is now possible to use a implicit EM
FEMPIC method while satisfying Gauss laws to machine precision. This enables
choosing time step sizes dictated by physics as opposed to geometry. In this
paper, we take this a step further. For devices characterized by a narrowband
high frequency response, choosing a time-step size based on the highest
frequency of interest is considerably expensive. In this paper, we use methods
derived from envelope tracking to construct an EM-FEMPIC method that
analytically provides for the high-frequency oscillations of the system,
allowing for analysis at considerable coarser time-step sizes even in the
presence of non-linear effects from active media such as plasmas.
Consequentially, we demonstrate how the pointwise metric used for measuring
satisfaction of Gauss Laws breaks down when prescribing analytical fast fields
and provide a thorough analysis of how charge conservation can be measured.
Through a number of examples, we demonstrate that the proposed approach retains
the accuracy the regular scheme while requiring far fewer time steps.
