Publisher Summary This chapter describes the self-imaging phenomenon and its applications. The self-imaging phenomenon requires a highly spatially coherent illumination. It disappears when the lateral dimensions of the light source are increased. When the source is made spatially periodic and is placed at the proper distance in front of the periodic structure, a fringe pattern is formed in the space behind the structure. The chapter discusses the theoretical and applicational aspects of the self-imaging phenomenon—that is, the property of the Fresnel diffraction field of some objects illuminated by a spatially coherent light beam. The applications of self-imaging are summarized in four main groups—namely, (1) image processing and synthesis, (2) technology of optical elements, (3) optical testing, and (4) optical metrology. The chapter describes the double diffraction systems using spatially incoherent illumination. The first periodic structure plays the role of a periodic source composed of a multiple of mutually incoherent slits. Depending on whether the periods of two periodic structures are equal, the Lau or the generalized Lau effect is discussed. Various applications of incoherent double-grating systems are described in the fields of optical testing, image processing, and optical metrology. After examining some cases of coherent and incoherent illumination, the general issue of spatial periodicities of optical fields and its relevance to the replication of partially coherent fields in space is discussed.

I The Self-Imaging Phenomenon and its Applications

Attention is drawn to the changes in the Fraunhofer diffraction patterns encountered when the planar objects become titled with respect to the optical axis. The differences between the far-field images observed in the well-known untilted case and the one investigated are shown and discussed using examples of familiar objects, such as square or rectangular apertures and linear diffraction gratings.

Fraunhofer Diffraction Patterns of Titled Planar Objects

Linear space-variant optical data processing

Explicit expressions for Fraunhofer diffraction patterns (spectra) of absolutely black bodies of constant thickness have been derived from the model of equivalent diaphragms in the Kirchhoff–Fresnel approximation. The expressions can be interpreted as results of interference between generalized point sources with corresponding amplitudes and Fresnel radiation diagrams located at boundary points of the object. Approximation of the Fresnel radiation diagrams by elementary functions is suggested for analytical estimation. For essential simplification of calculations, the quasi-geometrical approach has been developed, according to which the light diffraction by faces of the object is taken into account and propagation of diffracted waves in free space is described in the geometrical-optics approximation. The extension of objects is shown to result in Fresnel modulation of their spectra and also in disturbance of the equidistant position of diffraction minima. There is agreement between theoretical and experimental results.

Fraunhofer diffraction by volumetric bodies of constant thickness

In this chapter we have analyzed the basic characteristics of incoherent optical processing, considering three major classes of systems: (1) systems that rely on diffraction; (2) systems that rely on plane-to-plane imaging in the geometrical optics sense; and (3) systems that rely on diffractionless geometrical optics “shadow casting” for their operation. Incoherent systems are often characterized by a redundancy and immunity to noise not associated with coherent optical systems. However, the non-negative real nature of the information-bearing irradiance distributions precludes direct implementation of incoherent systems in many signal processing applications, and various tricks must be employed. Dynamic range limitations with incoherent systems are an area of active study, and the relative advantages of incoherent systems over coherent systems are not known conclusively.

Incoherent Optical Processing

The possibility of applying an ordinary quasi-geometrical technique for Fraunhofer diffraction calculations on absolutely absorbing three-dimensional (3-D) bodies of constant thickness is investigated. It is shown that such an application can lead to results that are inadequate for finding the physical diffraction pattern of 3-D bodies. A modified version of the technique is suggested that, to a greater degree, takes into account secondary diffraction and thus permits a more exact presentation of characteristic features of the light diffracted by 3-D bodies. Some examples of this approach applied to the light-diffraction analysis of simple 3-D bodies are given. It is shown experimentally and with calculations that this approach permits description of the diffraction effects of 3-D bodies of the class mentioned in rather simple form and with good accuracy.

Quasi-geometrical method for Fraunhofer diffraction calculations for three-dimensional bodies

A non coherent optical system has been proposed for performing transforms equivalent to the multiplication of three matrices. The possibility of using such a system has been shown for a spectral analysis of images according to an arbitrary basis with separated variables and for a multichannel processing of signals. For the optical system characteristic parameters the size of the matrices under multiplication is of the order of 100 × 100.

S. V. Mikhlyaev

Papers

Quasi-geometrical method for Fraunhofer diffraction calculations for three-dimensional bodies

Non-Coherent Optical System for Processing of Images and Signals