  Supernova remnants (SNRs) are believed to accelerate particles up to high
energies through the mechanism of diffusive shock acceleration (DSA). Except
for direct plasma simulations, all modeling efforts must rely on a given form
of the diffusion coefficient, a key parameter that embodies the interactions of
energetic charged particles with the magnetic turbulence. The so-called Bohm
limit is commonly employed. In this paper we revisit the question of
acceleration at perpendicular shocks, by employing a realistic model of
perpendicular diffusion. Our coefficient reduces to a power-law in momentum for
low momenta (of index $\alpha$), but becomes independent of the particle
momentum at high momenta (reaching a constant value $\kappa_{\infty}$ above
some characteristic momentum $p_{\rm c}$). We first provide simple analytical
expressions of the maximum momentum that can be reached at a given time with
this coefficient. Then we perform time-dependent numerical simulations to
investigate the shape of the particle distribution that can be obtained when
the particle pressure back-reacts on the flow. We observe that, for a given
index $\alpha$ and injection level, the shock modifications are similar for
different possible values of $p_{\rm c}$, whereas the particle spectra differ
markedly. Of particular interest, low values of $p_{\rm c}$ tend to remove the
concavity once thought to be typical of non-linear DSA, and result in steep
spectra, as required by recent high-energy observations of Galactic SNRs.
