Is there a curve $C$ that connects $(0,1)$ to $(a,0)$ for some $a>0$, and, when reflected to $C'$ in the $x$-axis, the shape $S=C \cup C'$ has the property that each horizontal light ray entering $S$ from the left reflects off of $S$ (interpreted as a perfect mirror) in such a way that it never emerges, i.e. it never again crosses $x=0$?

For example, a straight line $C$ fails to be such a curve: