  The Sachs equations governing the evolution of the optical matrix of geodetic
WANDs (Weyl aligned null directions) are explicitly solved in n-dimensions in
several cases which are of interest in potential applications. This is then
used to study Einstein spacetimes of type III and N in the higher dimensional
Newman-Penrose formalism, considering both Kundt and expanding (possibly
twisting) solutions. In particular, the general dependence of the metric and of
the Weyl tensor on an affine parameter r is obtained in a closed form. This
allows us to characterize the peeling behaviour of the Weyl "physical"
components for large values of r, and thus to discuss, e.g., how the presence
of twist affects polarization modes, and qualitative differences between four
and higher dimensions. Further, the r-dependence of certain non-zero scalar
curvature invariants of expanding spacetimes is used to demonstrate that
curvature singularities may generically be present. As an illustration, several
explicit type N/III spacetimes that solve Einstein's vacuum equations (with a
possible cosmological constant) in higher dimensions are finally presented.
