We carry out a detailed analysis of angle-sensitive devices based on the critical-angle effect. We consider their use in measuring small angular deflections of a laser beam. We establish the diffraction limit to the sensitivity for optical-angle sensors based on reflection and transmission of a laser beam. We find that this limit is identical to that of the triangulation scheme when using a position-sensitive detector or the autocollimation scheme. We analyze the main proposals to date of optical-angle sensors based on the critical-angle effect, focusing on their maximum sensitivity and their polarization dependence in practical conditions. We propose and analyze theoretically a novel and simple angle-sensitive device for sensing optical-beam deflections with very low polarization dependence and a maximum sensitivity close to the diffraction limit when used with typical laser beams. We discuss the basic principles for designing this type of device, provide numerical results, and point out a convenient fabrication procedure.