  We call a set $K \subset {\mathbb R}^s$ with positive Lebesgue measure a {\it
spectral set} if $L^2(K)$ admits an exponential orthonormal basis. It was
conjectured that $K$ is a spectral set if and only if $K$ is a tile (Fuglede's
conjecture). Despite the conjecture was proved to be false on ${\mathbb R}^s$,
$s\geq 3$ ([T], [KM2]), it still poses challenging questions with additional
assumptions. In this paper, our additional assumption is self-similarity. We
study the spectral properties for the class of self-similar tiles $K$ in
${\mathbb R}$ that has a product structure on the associated digit sets. We
show that any strict product-form tiles and the associated modulo product-form
tiles are spectral sets. As for the converse question, we give a pilot study
for the self-similar set $K$ generated by arbitrary digit sets with four
elements. We investigate the zeros of its Fourier transform due to the
orthogonality, and verify Fuglede's conjecture for this special case.
