  Plasma lenses in the host galaxies of fast radio bursts (FRBs) can strongly
modulate FRB amplitudes for a wide range of distances, including the $\sim $
Gpc distance of the repeater FRB121102. To produce caustics, the lens'
dispersion-measure depth (${\rm DM}_{\ell}$), scale size ($a$), and distance
from the source ($d_{\rm sl}$) must satisfy ${\rm DM}_{\ell} d_{\rm sl} / a^2
\gtrsim 0.65~ {\rm pc^2 \ AU^{-2} \ cm^{-3}}$. Caustics produce strong
magnifications ($\lesssim 10^2$) on short time scales ($\sim$ hours to days and
perhaps shorter) along with narrow, epoch dependent spectral peaks (0.1 to
1~GHz). However, strong suppression also occurs in long-duration ($\sim$
months) troughs. For geometries that produce multiple images, the resulting
burst components will arrive differentially by $< 1~\mu$s to tens of ms and
they will show different apparent dispersion measures, $\delta{\rm DM}_{\rm
apparent} \sim 1$ pc cm$^{-3}$. Arrival time perturbations may mask any
underlying periodicity with period $\lesssim 1$ s. When arrival times differ by
less than the burst width, interference effects in dynamic spectra are
expected. Strong lensing requires source sizes smaller than $({\rm
Fresnel~scale)^2} / a$, which can be satisfied by compact objects such as
neutron star magnetospheres but not by AGNs. Much of the phenomenology of the
repeating fast radio burst source FRB121102 is similar to lensing effects. The
overall picture can be tested by obtaining wideband spectra of bursts (from
$<1$ to 10 GHz and possibly higher), which can also be used to characterize the
plasma environment near FRB sources. A rich variety of phenomena is expected
from an ensemble of lenses near the FRB source. We discuss constraints on
densities, magnetic fields, and locations of plasma lenses related to
requirements for lensing to occur.

High Energy Astrophysical Phenomena