Magnetic multilayers of $(SrRuO_3)_m(SrMnO_3)_n$ were grown artificially using the pulsed laser deposition technique on (001)-oriented $SrTiO_3$ substrates. The state of strain at the interfaces and the structural coherency are studied in details utilizing asymmetrical $X$-ray diffraction and the $\sin ^2\psi $ method. First, the evolution of the lattice parameters, the crystallinity and the epitaxy of the films are evaluated as a function of the number of $SrMnO_3$ unit cells using $X$-rays diffraction and transmission electron microscopy. Second, our results on the stress indicate that the $SrRuO_3/SrMnO_3$ superlattices show a larger residual strain as compared to the single layer film of $SrRuO_3$. This suggests that the lattice stiffening from interfacial strain and inhibiting the dislocation by composition modulation. Finally, these results bring insights on the interfacial stress measurements of oxide multilayers that can be used to control the physical properties at the level of the atomic scale.