  Context. Relativistic jets emerging from AGN cores transfer energy from the
core to their surrounding ISM/IGM. Because jets are observed to have finite
opening angles, one needs to quantify the role of conical versus cylindrical
jet propagation in this energy transfer. Aims. We use FR-II AGN jets parameter
with finite opening angles. We study the effect of the variation of the opening
angle on the dynamics and energy transfer of the jet. We also point out how the
characteristics of this external medium, such as its density profile, play a
role in the dynamics. Methods. This study exploits our parallel AMR code
MPI-AMRVAC with its special relativistic hydrodynamic model, incorporating an
equation of state with varying effective polytropic index. We studied mildly
under-dense jets up to opening angles of 10 degrees, at Lorentz factors of
about 10, inspired by observations. Instantaneous quantification of the various
ISM volumes and their energy content allows one to quantify the role of mixing
versus shock-heated cocoon regions over the time intervals. Results. We show
that a wider opening angle jet results in a faster deceleration of the jet and
leads to a wider cocoon dominated by Kelvin-Helmholtz and Rayleigh-Taylor
instabilities. The energy transfer mainly occurs in the shocked ISM region by
both the frontal bow shock and cocoon-traversing shock waves, in a roughly 3 to
1 ratio to the energy transfer of the mixing zone, for a 5 degree opening angle
jet. A rarefaction wave induces a dynamically formed layered structure of the
jet beam. Conclusions. Finite opening angle jets can efficiently transfer
significant fractions (25 % up to 70 %) of their injected energy over a growing
region of shocked ISM matter. The role of the ISM stratification is prominent
for determining the overall volume that is affected by relativistic jet
injection.
