  Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension
$2n+1, n \ge 1$ with a transversal CR $S^1$ action on $X$. We establish an
asymptotic expansion for the $m$-th Fourier component of the Szeg\H{o} kernel
function as $m\rightarrow\infty$, where the expansion involves a contribution
in terms of a distance function from lower dimensional strata of the $S^1$
action. We also obtain explicit formulas for the first three coefficients of
the expansion.
