  Using the ``composite harmonic mapping method," we construct exact solutions
for cylindrically symmetric gravitational and electromagnetic waves within the
Einstein-Maxwell system, focusing on the conversion dynamics between these
types of waves. In this approach, we employs two types of geodesic surfaces in
${\mathbb H}^{2}_{C}$: (a) the complex line and (b) the totally real Lagrangian
plane, applied to two different vacuum seed solutions: (i) a vacuum solution
previously utilized in our studies and (ii) the solitonic vacuum solution
constructed previously by Economou and Tsoubelis. We study three scenarios:
case (a) with seeds (i) and (ii), and case (b) with seed (ii). In all cases (a)
and (b), solutions demonstrate notable mode conversions near the symmetric
axis. In case (a) with seed (i) or seed (ii), we show that any change in the
occupancy of the gravitational or electromagnetic mode relative to the C-energy
near the axis always reverts to its initial state once the wave moves away from
the axis. Particularly in case (b) with seed (ii), nontrivial conversions occur
even when the wave moves away from the axis. In this case, the amplification
factors of electromagnetic modes range from an upper limit of approximately
$2.4$ to a lower limit of about $0.4$, when comparing the contributions of
electromagnetic mode to C-energy at past and future null infinities.
