  We discuss a novel method for generating octave-spanning supercontinua and
few-cycle pulses in the important mid-IR wavelength range. The technique relies
on strongly phase-mismatched cascaded second-harmonic generation (SHG) in
mid-IR nonlinear frequency conversion crystals. Importantly we here investigate
the so-called noncritical SHG case, where no phase matching can be achieved but
as a compensation the largest quadratic nonlinearities are exploited. A
self-defocusing temporal soliton can be excited if the cascading nonlinearity
is larger than the competing material self-focusing nonlinearity, and we define
a suitable figure of merit to screen a wide range of mid-IR dielectric and
semiconductor materials with large effective second-order nonlinearities
$d_{\rm eff}$. The best candidates have simultaneously a large bandgap and a
large $d_{\rm eff}$. We show selected realistic numerical examples using one of
the promising crystals: in one case soliton pulse compression from 50 fs to 15
fs (1.5 cycles) at $3.0\mic$ is achieved, and at the same time a 3-cycle
dispersive wave at $5.0\mic$ is formed that can be isolated using a long-pass
filter. In another example we show that extremely broadband supercontinua can
form spanning the near-IR to the end of the mid-IR (nearly 4 octaves).
