  We analyze the effect of pivotal structures (on a 2-category) on the planar
algebra associated to a 1-cell as in \cite{Gho08} and come up with the notion
of {\em perturbations of planar algebras by weights} (a concept that appeared
earlier in Michael Burns' thesis \cite{Bur03}); we establish a one-to-one
correspondence between weights and pivotal structures. Using the construction
of \cite{Gho08}, to each bifinite bimodule over $II_1$-factors, we associate a
{\em bimodule planar algebra} in such a way that extremality of the bimodule
corresponds to sphericality of the planar algebra. As a consequence of this, we
reproduce an extension of Jones' theorem (\cite{Jon}) (of associating
`subfactor planar algebras' to extremal subfactors). Conversely, given a
bimodule planar algebra, we construct a bifinite bimodule whose associated
bimodule planar algebra is the one which we start with, using perturbations and
Jones-Walker-Shlyakhtenko-Kodiyalam-Sunder method of reconstructing an extremal
subfactor from a subfactor planar algebra. The perturbation technique helps us
to construct an example of a family of non-spherical planar algebras starting
from a particular spherical one; we also show that this family is associated to
a known family of subfactors constructed by Jones.
