In this paper, we introduced double cyclic codes over $R^r\times R^s$, where $R={\mathbb{Z}}_2+u{\mathbb{Z}}_2=\{0,1,u,1+u\}$ is the ring with four elements and $u^2=0$. We first determined the generator polynomials of $R$-double cyclic codes for odd integers $r$ and $s$, then gave the generators of duals of free double cyclic codes over $R^r\times R^s$. By defining a linear Gray map, we looked at the binary images of $R$-double cyclic codes and gave several examples of optimal parameter binary linear codes obtained from $R$-double cyclic codes. Moreover, we studied self-dual $R$-double cyclic codes and presented an example of a self-dual $R$-double cyclic code.