  We study cascaded quadratic soliton compressors and address the physical
mechanisms that limit the compression. A nonlocal model is derived, and the
nonlocal response is shown to have an additional oscillatory component in the
nonstationary regime when the group-velocity mismatch (GVM) is strong. This
inhibits efficient compression. Raman-like perturbations from the cascaded
nonlinearity, competing cubic nonlinearities, higher-order dispersion, and
soliton energy may also limit compression, and through realistic numerical
simulations we point out when each factor becomes important. We find that it is
theoretically possible to reach the single-cycle regime by compressing
high-energy fs pulses for wavelengths $\lambda=1.0-1.3 \mu{\rm m}$ in a
$\beta$-barium-borate crystal, and it requires that the system is in the
stationary regime, where the phase mismatch is large enough to overcome the
detrimental GVM effects. However, the simulations show that reaching
single-cycle duration is ultimately inhibited by competing cubic nonlinearities
as well as dispersive waves, that only show up when taking higher-order
dispersion into account.
