  We present optical vector network analysis (OVNA) of an isotopically purified
$^{166}$Er$^{3+}$:$^7$LiYF$_4$ crystal. The OVNA method is based on generation
and detection of modulated optical sideband by using a radio-frequency vector
network analyzer. This technique is widely used in the field of microwave
photonics for the characterization of optical responses of optical devices such
as filters and high-Q resonators. However, dense solid-state atomic ensembles
induce a large phase shift on one of the optical sidebands which results in the
appearance of extra features on the measured transmission response. We present
a simple theoretical model which accurately describes the observed spectra and
helps to reconstruct the absorption profile of a solid-state atomic ensemble as
well as corresponding change of the refractive index in the vicinity of atomic
resonances.
