  The widely-adopted proximity-force approximation (PFA) to estimate normal
Casimir forces is known to be asymptotically exact at vanishing separations. In
this letter, we propose a correction to the PFA, which is sufficiently accurate
in predicting displacement-induced lateral Casimir forces between a sphere and
a grating, for separation-to-radius ratio up to 0.5, far beyond the limit
within which the application of PFA is previously restricted. Our result allows
convenient estimation of Casimir interactions and thus shall be useful in
relevant experimental and engineering Casimir applications. We also study the
PFA for gradient gratings, and we find that the inhomogeneity-induced lateral
Casimir force is beyond the corrected PFA.
