The interferometric antenna response to gravitational-wave excitation is studied with the Fermi Normal Coordinate (FNC) reference system, with the limit that the gravitational wavelength must be much larger than the interferometric arm length. An optical configuration of the antenna, quite similar to the one generally considered in the long-baseline antenna projects, has been studied, i.e., Fabry-Perot optical cavities in the interferometer arms and reflectors at the input and the output of the interferometer for the purpose of recycling both the laser power and the output signals. An exact computation of the antenna response is given in a form that is also suitable to provide directly the responses for simplier optical configurations without the power or the signal recycling. Furthermore the response of the antenna for the narrow-band detection mode is also calculated. The results obtained in the FNC gauge at rest with the antenna are consistent with the ones given in the literature that were computed in the transverse traceless gauge at rest with the gravitational radiation.