We show that in superlattices with alternating superconducting (S) and ferromagnetic (F) layers the spontaneous magnetic field induced in the superconducting layers due to the electromagnetic proximity effect becomes dramatically enhanced compared to the previously studied S/F bilayers. The effect reveals itself for the in-plane orientation of the magnetic moments, both for ferromagnetic and antiferromagnetic ordering of the moments in the F layers. In the finite-size samples the magnetic field decays from the sample surface towards the bulk of the structure and the decay length strongly depends on the relative orientation of the sample surface, the layers planes and magnetic moments in the F layers. The obtained results provide additional insights into experimental data on the neutron scattering in Nb/Gd superlattices.