  We discuss production of $W^+ W^-$ pairs in proton-proton collisions induced
by two-photon fusion including, for a first time, transverse momenta of
incoming photons. The unintegrated inelastic fluxes (related to proton
dissociation) of photons are calculated based on modern parametrizations of
deep inelastic structure functions in a broad range of their arguments ($x$ and
$Q^2$). In our approach we can get separate contributions of different $W$
helicities states. Several one- and two-dimensional differential distributions
are shown and discussed. The present results are compared to the results of
previous calculations within collinear factorization approach. Similar results
are found except of some observables such as e.g. transverse momentum of the
pair of $W^+$ and $W^-$. We find large contributions to the cross section from
the region of large photon virtualities. We show decomposition of the total
cross section as well as invariant mass distribution into polarisation states
of both W bosons. The role of the longitudinal $F_L$ structure function is
quantified. Its inclusion leads to a 4-5 % decrease of the cross section,
almost independent of $M_{WW}$.
