Energy considerations, propagation in random medium and imaging in scalar coherence theory

Thesis (Ph.D.)--Boston UniversityAfter a review of scalar coherence theory the infinite-time averages of coherence theory are interpreted in terms of the conserved quantities of any scalar wave field and these averages are related to the finite-time averages of physical measurements. In particular it is shown that the infinite-time average of the square of the wave field (intensity) is proportional to the average energy flux under many, but not all, circumstances. By application of the energy conservation law for the scalar wave field it is shown that the average energy flux through a plane surface depends not only upon the intensity but also upon the correlation of the field. An explicit example of this dependence is given. Further the assumptions which underlie the association of infinite-time averages with actual physical measurements are given explicitly. After a review of ensemble coherence theory, an ergodic theorem which relates the time and ensemble average coherence theories is presented. In particular it is shown that if the boundary conditions are ergodic (i.e., the ensemble averages on the boundary are equal to the time averages), then the solutions are also ergodic. In Chapter VI the investigation is turned from the formal properties of coherence theory to an application of the theory to a particular problem in wave motion - the propagation of waves in a random medium. Random medium here means any region of space for which the speed of the waves varies randomly in both space and time. By invoking an ergodic hypothesis the time and space varying medium is replaced by an ensemble of media which vary only in space. For such media it is shown that the ensemble average behavior depends only upon the ensemble averaged, two-point cross correlation of the fluctuations in the index of refraction of the media. Further, an approximate solution to the problem of the propagation of a wave beam of finite cross -section and divergence (spread) through a slab of Gaussianly correlated random medium is given. The principle conclusions are that the slab produces mainly forward scattering and that the scattered wave is highly coherent. In Chapter VII a wave theory of imaging is developed from the equations of motion for the coherence. From this theory explicit representations for imaging in the cases of coherent and incoherent illumination are derived. These representations show that imaging with coherent illumination is basically nonlinear in terms of what is observable in the object plane and observable in the image plane. In Chapter VIII various consequences of the nonlinear equation governing imaging with coherent illumination are exhibited. Lacking a general theory of nonlinear equations, the properties of coherent imaging cannot be summarized by giving a Green's function or transfer function. Instead the nature of the images of simple geometrical objects are given in some detail. From examples of coherent imaging it is concluded that the two most striking features are that the images of edges "ring" and appear to be shifted in relation to the same images under incoherent illumination. By "ring" it is meant that the image of an edge is not a monotonic change in intensity, but rather, that interference fringes occur in the neighborhood of the location of the edge. Further, coherent imaging is linearized by restricting the object to be of low contrast. Having linearized coherent imaging, it is directly compared with incoherent imaging. Finally, a particular problem in optics is analyzed - that of the sparkle of laser light. It is shown that all the observed properties of the sparkle can be explained by considering the nature of the image of a coherently illuminated, rough surface. By a rough surface it is meant any surface which deviates randomly from a plane (or other simple geometrical surface that could be constructed to be coincident with a wavefront of the illumination). It is shown that the image of a coherently illuminated rough surface is as if incoherent illumination were used - edge ringing and shifting are not present - but that the image is modified by random fluctuations in intensity. These random fluctuations give the image a speckled appearance. It is shown that this speckling is Rayleigh distributed and the dependence of the statistical parameters upon the nature of the imaging system (lens) is exhibited