For a system that consists of a transparent bilayer on an absorbing substrate of known optical properties, we determine all possible combinations of the thicknesses of the two layers such that the differential reflection phase shift between the p and s polarizations Delta = +/-90 degrees for monochromatic light incident al a given angle phi. A ZnS-ThF4 bilayer on a Ag substrate at 3.39 mu m wavelength is considered as an example of a high-reflectance system. Constant principal-angle contours are presented in the plane of the normalized thicknesses of the two films. Four distinct bilayers on Ag make Delta = 90 degrees and equalize the reflectances for the p and s polarizations at phi = 65 degrees, hence realize an ideal quarter-wave retarder (QWR). The QWR with the thinnest bilayer has the lowest sensitivity to incidence angle and wavelength shifts. A second example is that of a Si3N4-SiO2 bilayer on Si at 633 nm wavelength. The results in this case are of interest in relation to the four-detector photopolarimeter, and also for principal-angle ellipsometry on this technologically important material system.