Serial transform optical correlator design principles

Optical correlators such as the 4f and VanderLugt optical systems have been an active area of research for many years; we refer to these types of optical system collectively as serial transform correlators (STCs). Despite being well known, misconceptions regarding the design of STCs are not uncommon. We show, for example, that one correlator configuration reported to suffer from a phase curvature problem in fact works correctly. We present and prove a simple set of rules to follow in the selection of a STC design that does not contain unnecessary constraints and that makes it easier to identify permissible optical systems. As examples, we discuss three representative types of correlator configuration, one of which is discussed in detail to highlight its practical advantages. A novel proof of the design rules is presented that does not depend on details such as what lenses or combination of lenses are used, what their focal lengths are, or what their locations are within the optical system. We also present a conventional Fourier optics proof.