  We lay a theoretical foundation in the observation of gravitational waves
(GWs) by electromagnetic waves (EMWs) performing a full electromagnetic
analysis without any optical approximation. For that, the perturbation of plane
EMWs is obtained by solving the perturbed Maxwell equation with GWs in the
Minkowski background spacetime. In a GW detector using the EMWs, we propose to
measure the electromagnetic invariants that are independent of the motion of
the EMW receiver and whose perturbations are gauge-invariant. Imposing a
physical boundary condition at EMW emitters, we have analytic results for the
perturbations of invariants that can be measured in the EMW receiver. Finally,
we show antenna patterns of the detector with monochromatic plane GWs and EMWs.
