Interferometric electro-optical signal processors with partially coherent illumination

Optical image processing can derive significant benefits from the introduction of interferometric methods. The use of a shearing interferometer to yield the cosine Fourier transform of an object illuminated by spatially incoherent light is well known.'2 Other applications of the same interferometer include the sine transform'and the Hartley transform (HT) in coherent6'7 and incoherent8 illumination. In this paper we consider some additional properties of both coherent and incoherent interferometers for signal processing. Since these processing architectures usually contain Fourier transformations (FT's), we refer to them as Fourier interferometers (FI's). A generalized model of the FI is discussed in Section 2. One important attribute of the HT is that it yields a real transform for a real input function. However, this is not necessarily the case for an arbitrary complex input function. We show in the analysis of Section 3 that system modifications are possible for generating real transforms also for complex functions. The joint transform correlator9 (JTC) is a convenient system for obtaining the cross correlation between two functions without the need for holographic filters and complicated alignment procedures. While the conventional JTC required fully coherent illumination, we demonstrate in Section 5 that a JTC implemented by the FI can be operated under spatially incoherent illumination. In the proposed JTC the FT of the input objects is performed with incoherent illumination, but the processing of the joint spectrogram to obtain the correlation uses coherent light. The preliminary experimental results confirm the theoretical predictions. In Section 6 we propose and demonstrate a partially incoherent correlator characterized by Fourier transforming an incoherently illuminated object, multiplying the transform by a filter, and performing an additional FT with coherent illumination. This configuration is useful in a situation in whch it is preferable to illuminate coherently the spatial frequency plane rather than to convert the input image into a coherently illuminated image.