The fully relativistic spin-polarized screened Korringa-Kohn-Rostoker method is used to evaluate the electronic and magnetic structure as well as the optical conductivity of (Co_mIr_m)_n superstructures on Ir(111). By mapping the microscopic optical conductivity tensor onto the macroscopic permittivity tensor by using the so-called 2 x 2 matrix technique the surface reflectivity matrices for these systems are then calculated, from which in turn the Kerr rotation and ellipticity angles can be determined. It is found (i) that when varying at a given value of m the number of repetitions n, these angles are linearly proportional to the total magnetic moment, and (ii) that at a frequency of about 3.8 eV the Kerr rotation angles have the largest value, the corresponding maximum being mainly caused by the Ir spacer layers. The optical properties of the free surface of Ir(111), which is considered to check the applied theoretical schemes, turn out to be in good agreement with existing experimental data.