Here, we describe in detail a procedure for the numerical design of planar focusing mirrors based on monolithic high contrast gratings. We put a special emphasis on the reconstruction of the hyperbolic phase of these mirrors and we conclude that the phase does not have to be perfectly mimicked to obtain a focusing reflector. We consider here the grating mirrors that focus light not in the air but in the GaAs substrate and we compare them with conventional parabolic reflectors of corresponding dimensions. The light intensity at the focal point of the focusing grating mirrors was found to be comparable to that of the parabolic reflector. Moreover, the reflectivity of the focusing grating mirrors is almost as high as that of parabolic mirrors covered with an additional reflecting structure, if the ratio of the reflector width to the focal length is less than 0.6. Planar focusing grating mirrors offer a good alternative to parabolic mirrors, especially considering the complexity of fabricating three-dimensional structures compared to planar structures.