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.
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