Chapter 1
Introduction

Optics, as a field of science, is in its third millennium of life; yet in spite of its age, it remains remarkably vigorous and youthful. During the middle of the twentieth century, various events and discoveries gave new life, energy, and richness to the field. Especially important in this regard were (i) the introduction of the concepts and tools of Fourier analysis and communications theory into optics, primarily in the late 1940s and throughout the 1950s; (ii) the invention of the laser in late 1950s and its commercialization starting in the early 1960s; (iii) the origin of the field of nonlinear optics in the 1960s; (iv) the invention of the low-loss optical fiber in the early 1970s and the revolution in optical communications that followed; and (v) the rise of the young fields of nanophotonics and biophotonics. It is the thesis of this book that in parallel with these many advances, another important change has also taken place gradually but with an accelerating pace, namely, the infusion of statistical concepts and methods of analysis into the field of optics. It is to the role of such concepts in optics that this book is devoted.

The field we shall call "statistical optics" has a considerable history of its own. Many fundamental statistical problems were solved in the late nineteenth century and applied to acoustics and optics by Lord Rayleigh. The need for statistical methods in optics increased dramatically with the discovery of the quantized nature of light and, particularly, with the statistical interpretation of quantum mechanics introduced by Max Born. The introduction by E. Wolf in 1954 of an elegant and broad framework for considering the coherence properties of waves laid a foundation upon which many of the important statistical problems in optics could be treated in a unified way. Also worth mentioning is the semiclassical theory of light detection, pioneered by L. Mandel, which tied together, in a comparatively simple way, knowledge of the statistical fluctuations of classical wave quantities (fields, intensities) and fluctuations associated with the interaction of light and matter. This history is far from complete but is dealt with in more detail in the individual chapters that follow.

1.1 Deterministic Versus Statistical Phenomena and Models

In the normal course of events, a student of physics or engineering first encounters optics in an entirely deterministic framework. Physical quantities are represented by mathematical functions that are either completely specified in advance or are assumed precisely measurable. These physical quantities are subjected to well-defined transformations that modify their form in perfectly predictable ways. For example, if a monochromatic light wave with a known complex field distribution is incident on a transparent aperture in an otherwise opaque screen, the resulting complex field distribution some distance behind the screen can be calculated using the well-established diffraction formulas of wave optics. In this approach, inaccuracies in the results arise only due to inaccuracies of the deterministic models used to describe the diffraction process.

The students emerging from such an introductory course may feel confident that they have grasped the basic physical concepts and laws and are ready to find a precise answer to almost any problem that comes their way. To be sure, they have probably been warned that there are certain problems, arising particularly in the detection of weak light waves, for which a statistical approach is required. But a statistical approach to problem solving appears at first glance to be a "second class" approach, for statistics is generally used when we lack sufficient information to carry out the aesthetically more pleasing "exact" solution. The problem may be inherently too complex to be solved analytically or numerically, or the boundary conditions may be poorly defined. Surely, the preferred way to solve a problem must be the deterministic way, with statistics entering only as a sign of our weakness or limitations. Partially as a consequence of this viewpoint, the subject of statistical optics is usually left for the more advanced students, particularly those with a mathematical flair.

Although the origins of the above viewpoint are quite clear and understandable, the conclusions reached regarding the relative merits of deterministic and statistical analysis are very greatly in error, for several important reasons. First, it is difficult, if not impossible, to conceive of a real engineering problem in optics that does not contain some element of uncertainty requiring statistical analysis. Even the lens designer, who traces rays through application of precise physical laws accepted for centuries, must ultimately worry about quality control! Thus statistics is certainly not a subject to be left primarily to those more interested in mathematics than in physics and engineering.

Furthermore, the view that the use of statistics is an admission of one's limitations and thus should be avoided is based on too narrow a view of the nature of statistical phenomena. Experimental evidence indicates, and indeed the great majority of physicists believe, that the interaction of light and matter is fundamentally a statistical phenomenon, which in principle cannot be predicted with perfect precision in advance. Thus statistical phenomena play a role of the greatest importance in the world around us, independent of our particular mental capabilities or limitations.

Finally, in defense of statistical analysis, we must say that, whereas both deterministic and statistical approaches to problem solving require the construction of mathematical models of physical phenomena, the models constructed for statistical analysis are inherently more general and flexible. Indeed, they invariably contain the deterministic model as a special case! For a statistical model to be accurate and useful, it should fully incorporate the current state of our knowledge regarding the physical parameters of concern. Our solutions to statistical problems will be no more accurate than the models we use to describe both the physical laws involved and the state of knowledge or ignorance.

The statistical approach is indeed somewhat more complex than the deterministic approach, for it requires knowledge of the elements of probability theory. In the long run, however, statistical models are far more powerful and useful than deterministic models in solving physical problems of genuine practical interest. Hopefully, the reader will agree with this viewpoint by the time this book has been digested.

1.2 Statistical Phenomena in Optics

Statistical phenomena are so plentiful in optics that there is no difficulty in compiling a long list of examples. Because of the wide variety of these problems, it is difficult to find a general scheme for classifying them. Here we attempt to identify several broad aspects of optics that require statistical treatment. These aspects are conveniently discussed in the context of an optical imaging problem.

Most optical imaging problems are of the following type. Nature assumes some particular state (e.g., a certain collection of atoms and/or molecules in a distant region of space, a certain distribution of reflectance over terrain of unknown characteristics, or a certain distribution of transmittance in a sample of interest). By operating on optical waves that arise as a consequence of this state of Nature, we wish to deduce exactly what that state is.

Statistics is involved in this task in a wide variety of ways, as can be discovered by reference to Fig. 1.1.

Figure 1.1 An optical imaging system.

First, and most fundamentally, the state of Nature is known to us a priori only in a statistical sense. If it were known exactly, there would be no need for any measurement in the first place. Thus the state of Nature is random, and in order to properly assess the performance of the system, we must have a statistical model, ideally representing the set of possible states, together with associated probabilities of the occurrence of those states. Usually, a less complete description of the statistical properties of the object will suffice.

Our measurement system operates not on the state of Nature per se, but rather on, an optical representation of that state (e.g., radiated light, transmitted light, or reflected light). The representation of the state of Nature by an optical wave has statistical attributes itself, primarily as a result of the statistical or random properties of all light waves. Because of fundamentally statistical attributes of the mechanisms that generate light, all optical sources produce radiation that is to some degree random in its properties. At one extreme, we have the chaotic and unordered emission of light by a thermal source, such as an incandescent lamp; at the other extreme, we have the comparatively ordered emission of light by a continuous-wave (CW) gas laser. In the latter case, the light comes close to containing only a single frequency and traveling in a single direction. Nonetheless, any real laser emits light with statistical properties, including random fluctuations of both amplitude and phase of the radiation. Statistical fluctuations of light are of great importance in many optical experiments and indeed play a central role in the character of the image produced by the system depicted in Fig. 1.1.

After interacting with the state of Nature, the radiation travels through an intervening medium until it reaches the focusing optics. The parameters of that...