  We consider polynomials $p_n^{\omega}(x)$ that are orthogonal with respect to
the oscillatory weight $w(x)=e^{i\omega x}$ on $[-1,1]$, where $\omega>0$ is a
real parameter. A first analysis of $p_n^{\omega}(x)$ for large values of
$\omega$ was carried out in connection with complex Gaussian quadrature rules
with uniform good properties in $\omega$. In this contribution we study the
existence, asymptotic behavior and asymptotic distribution of the roots of
$p_n^{\omega}(x)$ in the complex plane as $n\to\infty$. The parameter $\omega$
grows with $n$ linearly. The tools used are logarithmic potential theory and
the $S$-property, together with the Riemann--Hilbert formulation and the
Deift--Zhou steepest descent method.
