  Frequency-dependent brightness fluctuations of radio sources, the so-called
extreme scattering events (ESEs), have been observed over the last three
decades. They are caused by Galactic plasma structures whose geometry and
origin are still poorly understood. In this paper, we construct axisymmentric
two-dimensional (2D) column density profiles for the plasma lens and explore
the resulting ESEs for both point-like and extended sources. A quantity that
becomes relevant is the impact parameter $b$, namely the distance of the
observer's path from the lens symmetry axis. We demonstrate its effects on the
shape of ESE light curves and use it for a phenomenological classification of
ESEs into four main types. Three of them are unique outcomes of the 2D model
and do not show a characteristic U-shaped dip in the light curve, which has
been traditionally used as an identification means of ESEs. We apply our model
to five well-studied ESEs and show that elongated plasma tubes or
quasi-spherical clouds are favoured over plasma sheets for four of them, while
the remaining one is compatible with both lens geometries.
