This paper considers the issue of optimality of fractional factorial experiments involving m factors each at two levels. The optimality criteria used here is the type 1 criteria, due to Cheng (1978), which include the D- and A-criteria. It is shown that if there exists an orthogonal array OA(n - ${ell}$, m, 2, 3), ${ell}=1,2$, then there exists an n-run type 1 optimal fractional factorial plan for a $2^m$ experiment under a model that includes the mean, all main effects and all two-factor interactions with a factor in common. These plans are obtained by adding any one run to an OA(n - 1,m, 2, 3) for $n{equiv}1$ (mod 8) and two specific runs to an OA(n - 2, m, 2, 3) for $n{equiv}2$ (mod 8).