Introduction

If this book were to follow historical order, the present chapter should have preceded the previous one, since lenses and mirrors were known and studied long before wave theory was understood. However, once we have grasped the elements of wave theory, it is much easier to appreciate the strengths and limitations of geometrical optics, so logically it is really more appropriate to put this chapter here. Essentially, geometrical optics, which considers light waves as rays which propagate along straight lines in uniform media and are related by Snell's law (§§2.7.2, 5.4.2) at interfaces, has a similar relationship to wave optics as classical mechanics does to quantum mechanics. For geometrical optics to be strictly true, it is important that the sizes of the elements we are dealing with be large compared with the wavelength λ. This means that we can neglect diffraction, which otherwise prevents the exact simultaneous specification of the positions and directions of rays on which geometrical optics is based. From the practical point of view, geometrical optics answers most questions about optical instruments extremely well and in a much simpler way than wave theory could do; it fails only in that it can not define the limits of performance such as resolving power, and does not work well for very small devices such as optical fibres. These will be dealt with by wave theory in Chapters 10 and 12.

The plan of the chapter is first to treat the classical ray theory of thin lens systems in the paraxial approximation.

Occurrence of diffraction

As we saw in Chapter 1, the wave theory of light was not at first generally accepted because light did not appear to have any obviously wave-like properties; for example, it did not bend round obstacles as water waves are clearly seen to do. The reason why this difficulty no longer prevents our acceptance of the wave theory is that we are now aware of the relative scales of the two sorts of waves: water waves are coarse and we can see that they only bend round obstacles that have dimensions of the same order of magnitude as the wavelength; larger objects merely stop the waves in the sense that the waves bending round the edge produce negligible effects. But the wavelength of light is about 5 × 10-7 (0.5 μm) and an object of about a hundred waves in size – sufficient to stop a light wave – is still very small by ordinary standards. Nevertheless some bending of the light waves round the edges of obstacles does occur and can be observed over a range of conditions. For particles of the order of a few wavelengths in size, no special apparatus is needed; for example, the water droplets that condense on a car window are surprisingly uniform in size and show beautiful halos round the street lights as the car passes by. For objects that are much larger, special apparatus is needed. The effects are called diffraction phenomena.

Classical dispersion theory

Many aspects of the interaction between radiation and matter can be described quite accurately by a classical theory in which the medium is represented by model atoms consisting of positive and negative parts bound by an attraction which depends linearly on their separation. Although quantum theory is necessary to calculate from first principles the magnitude of the parameters involved, in this chapter we shall show that many optical effects can be interpreted physically in terms of this model by the use of classical mechanics. In §13.5 we shall relax the restriction of linearity. Some of the quantum mechanical foundations will be discussed briefly in Chapter 14, but most are outside the scope of this book (see Yariv, 1989; Loudon, 1983).

The term dispersion means the dependence of dielectric response (dielectric constant and refractive index) on frequency of the wave field. This will be the topic of the present section. Afterwards we shall see some of the applications of dielectric response to spatial effects.

The classical atom

Our classical picture of an atom consists of a massive positive nucleus surrounded by a light spherically-symmetrical cloud of electrons with an equal negative charge. We imagine the two as bound together by springs as in Fig. 13.1, so that in equilibrium the centres of mass and charge of the core and electron charge coincide. As a result the static atom has zero dipole moment.

There are two sorts of textbooks. On the one hand, there are works of reference to which students can turn for the clarification of some obscure point or for the intimate details of some important experiment. On the other hand, there are explanatory books which deal mainly with principles and which help in the understanding of the first type.

We have tried to produce a textbook of the second sort. It deals essentially with the principles of optics, but wherever possible we have emphasized the relevance of these principles to other branches of physics – hence the rather unusual title. We have omitted descriptions of many of the classical experiments in optics – such as Foucault's determination of the velocity of light – because they are now dealt with excellently in most school textbooks. In addition, we have tried not to duplicate approaches, and since we think that the graphical approach to Fraunhofer interference and diffraction problems is entirely covered by the complex-wave approach, we have not introduced the former.

For these reasons, it will be seen that the book will not serve as an introductory textbook, but we hope that it will be useful to university students at all levels. The earlier chapters are reasonably elementary, and it is hoped that by the time those chapters which involve a knowledge of vector calculus and complex-number theory are reached, the student will have acquired the necessary mathematics.

Introduction

As we saw in Chapter 5, electromagnetic waves in isotropic materials are transverse, their electric and magnetic field vectors E and H being normal to the direction of propagation k. The direction of E or rather, as we shall see later, the electric displacement field D, is called the polarization direction, and for any given direction of propagation there are two independent such vectors, which can be in any two mutually orthogonal directions normal to k. When the medium through which the wave travels is anisotropic, which means that its properties depend on orientation, the above statements meet with some restrictions. We shall see that the result of anisotropy in general is that the fields D and B remain transverse to k under all conditions, but E and H, no longer having to be parallel to D and B, are not necessarily transverse. Moreover, the two independent polarizations that propagate must now be chosen specifically with relation to the axes of the anisotropy. A further direct consequence of E and H no longer being necessarily transverse, is that the Poynting vector Π = E × H may not be parallel to the wave-vector k.

In this chapter, we shall first discuss the various types of polarized radiation that can propagate. We shall then go on to extend the theory of electromagnetic waves as described in Chapter 5 to take into account anisotropic media.

Importance of history

Why should a textbook on physics begin with history? Why not start with what is known now and refrain from all the distractions of out-of-date material? These questions would be justifiable if physics were a complete and finished subject; only the final state would then matter and the process of arrival at this state would be irrelevant. But physics is not such a subject, and optics in particular is very much alive and constantly changing. It is important for the student to understand the past as a guide to the future. To study only the present is equivalent to trying to draw a graph with only one point.

It can also be interesting and sometimes sobering to learn how some of the greatest ideas came about. By studying the past we can sometimes gain some insight – however slight – into the minds and methods of the great physicists. No textbook can, of course, reconstruct completely the workings of these minds, but even to glimpse some of the difficulties that they overcame is worthwhile. What seemed great problems to them may seem trivial to us merely because we now have generations of experience to guide us; or, more likely, we have hidden them by cloaking them with words. For example, to the end of his life Newton found the idea of ‘action at a distance’ repugnant in spite of the great use that he made of it; we now accept it as natural, but have we come any nearer than Newton to understanding it?

Introduction

Optical processors hold tremendous potential for processing many information channels in parallel, each at very high bandwidth, and in small volumes with low power consumption (VanderLugt, 1992). Major classes of operations that optics can perform include integral transformations such as Fourier transforms and correlations, and matrix operations, e.g. vector–matrix multiplication (Lee, 1987). Many, but not all, of these are made possible by the remarkable property that a ‘Fourier transform lens’ generates, at the back focal plane, the two-dimensional (2-D) Fourier transform of phase and amplitude information input at the front focal plane.

Initial feasibility demonstrations of optical signal processors are usually performed in the laboratory using existing components, but these often do not come close to fully exploiting the potential of the optical processor, and often disregard practical implementation issues such as: how much phase distortion is permissible in a critical lens, or what will be the effect of vibration or temperature changes in the field, and is it feasible to have a given dynamic range and information capacity in the same processor? Therefore the aim of most subsequent optical hardware development is to answer these questions and overcome any deficiencies.

Much effort has gone into the development of the critical active optical devices, such as infrared laser diodes, spatial light modulators (SLMs) and photodetector arrays, and new device concepts and improvements on existing devices continue to occur (Lee & VanderLugt, 1989).

Introduction

This chapter is dedicated to the evaluation of optical interconnects between electronic processors in multiprocessor systems. Each processor is fully electronic except for the incorporation of a number of photodetectors and optical signal transmitters (e.g., laser diodes, light emitting diodes (LEDs) or optical modulators).

The motivation for this analysis stems from fundamental advantages of optics as well as those of electronics. While electrons are charged fermions, subject to strong mutual interactions and strong reactions to other charged particles, photons are neutral bosons, virtually unaffected by mutual interactions and Coulomb forces. Thus, unlike electrons, multiple beams of photons can cross paths without significant interference. This property allows holographic interconnects to achieve a 3-D (three-dimensional) connection density with only 2-D optical elements. Similarly, photons can propagate through transparent materials without appreciable attenuation or power dissipation. Thus, neglecting speed of light delays, the speed of an optical link is limited only by the switching speed and capacitance of the transmitters and detectors. (For a 10 cm connection length, and a 50° hologram deflection angle, the speed-of-light delay is 0.5 ns.) Hence, the speed and power requirements of an optical interconnect are independent of the connection length. Since electrical very large scale integration (VLSI) connections have a switching energy directly proportional to the line length and an RC delay that grows quadratically with line length, for long enough communication links, optical connections will dissipate less power and provide faster data rate communication.

Introduction

When a scientist or engineer is provided with a given processing problem and asked whether optical processing techniques can solve it, how does he answer the question and approach the problem? To address this, he first considers the basic operations possible. They are often not a direct match to a new problem. Hence new algorithms arise to allow new operations to be performed (often on a new optical architecture). The final system is thus the result of an interaction between optical components, operations, architectures, and algorithms. This chapter describes these issues for the case of image processing. By limiting attention to this general application area, specific examples can be provided and the role for optical processing in most levels of computer vision can be presented. (Other chapters address other specific optical processing applications.)

Section 1.2 presents some general and personal philosophical remarks. These provide guidelines to be used when faced with a given data processing problem and to determine if optical processing has a role in all or part of a viable solution. Section 1.3 describes optical feature extractors. These are the simplest optical systems. This provides the reader with an introduction and summary of some of the many different operations possible on optical systems and how they are of use in image processing. A major application for these simple systems, product inspection, is described to allow a comparison of these optical systems and electronic systems to be made.

Section 1.4 addresses the optical correlator architecture, since it is one of the most powerful and most researched architectures. The key issues in such a system used for pattern recognition are noted.

The design of optical hardware for signal and image processors and in computing systems can be expected to become an increasingly important activity as optics moves from the laboratory to actual usage. Over the past decades a variety of concepts have been developed to exploit the capabilities of optics for ultrahigh-bandwidth and massively-parallel processing such as for analog front-end operations, image-processing and neural-net applications, high-bandwidth interconnection of electronic systems such as computers and microwave subsystems, execution of logic and other nonlinear operations for routing of optical interconnects, as well as for general reduction of hardware size, weight and power consumption. Original concept investigations usually identify theoretical advantages of performing various tasks with optics, but one then needs to address whether the advantages can be maintained as one addresses various new issues involving usage in a real-world environment; e.g., can demonstrations of optical processing speed and parallelism, optical routing, and communication bandwidth be retained in full-scale application? Addressing these questions very often requires an optical approach that employs different algorithmic/architectural approaches than conventional alternatives; hence, new architectures are devised to utilize maximally the strengths of optics. Finally, the capabilities of optical technology may dictate specific optical hardware designs.

It is therefore difficult to predict whether a specific optical hardware configuration will still possess the originally identified advantages. Although full-scale demonstrations can provide a direct answer, careful design and analyses must be performed to ensure an effective demonstration. The design and analysis stage can involve fundamental issues that are quite removed from the original optical-processing concept.

Introduction

This chapter discusses the issues involved in the design of a free-space digital optical logic system. The first section is an introduction to this field with an overview of the justification of pursuing this exciting and challenging course. The second section is a discussion of the basic characteristics of the devices that must be used in a system of this type with specific examples of the behavior of three of the most popular devices. Section 6.3 is a discussion of the optical and mechanical constraints involved in the design process. The fourth section is a design example using logic gates made from symmetric self electro-optic effect devices (S-SEEDs). The final section summarizes the likely future directions.

Background of digital optics

Two methods exist for communicating high bandwidth information: light and electricity. It is clear that the optimum choice for long distances (>1 km) is light; hence the adoption of fiber for telephony. It is equally clear that over short distances (<1 mm), the optimum choice is electricity hence its use for gate-to-gate interconnection in a chip. It has been shown that as the distance between the transmitter and receiver increases, the energy required by an electrical connection increases more rapidly than if optics is used (Miller, 1989). It is difficult to justify the use of optics over short distances and conversely, electrical connection is not ideal except over the shortest distances. The ultimate optimum choice for chip-to-chip and board-to-board interconnection is unclear.

Introduction

Optical linear algebra processors (OLAPs) perform numerical calculations such as matrix–vector multiplication by equating a particular property of light, normally its intensity level, to a number. Through various effects to be discussed in this section, uncertainty is introduced either through random fluctuations or the addition of a bias. The fluctuating intensity can now be considered a random variable for which a mean and standard deviation can be determined. After corrections for bias, the mean value can be taken as the correct level that relates to the expected numerical value while the nonzero standard deviation represents processor noise.

In the design of analog OLAPs such as matrix–vector and matrix–matrix processors, the effect of noise from both device and system sources has a major impact on the choice of system architecture and components. The ability of a particular system design to meet system performance specifications such as signal-to-noise ratio (SNR), dynamic range, and accuracy is directly connected to the noise properties of the system and its components. As an example, the choice of the spatial light modulator (SLM) has a significant impact (Casasent & Ghosh, 1985; Perle & Casasent, 1990; Taylor & Casasent, 1986; Batsell, Jong, Walkup & Krile, 1989a, 1990; Jong, Walkup & Krile, 1986). This chapter examines the effects of noise on these system specifications and their implications for OLAP design.

We begin with a theoretical analysis of a simple matrix–vector multiplier and then generalize the results for N cascaded matrix–vector multipliers.

Measurement methods based on the phenomenon known as coherent light speckle have become increasingly important in recent years. The development of electronic speckle pattern interferometry, in which the speckle patterns are acquired by a television system and then combined in a computer to create fringe patterns that can be displayed on a television monitor, has generated additional interest in speckle methods.

This chapter discusses the origins and nature of laser speckle, and it then goes on to describe the product of combining speckle fields in different ways. These notions are important to understanding the various methods by which speckle is employed in interferometry, as are described in subsequent chapters. Given the importance of electronic speckle techniques, the development assumes, where appropriate, that the recording of speckle irradiance is by means of an electronic detector rather than by a photographic emulsion.

Most of the concepts in this chapter have been treated exhaustively, lucidly, and creatively by Ennos (1975), Goodman (1975), Jones and Wykes (1983), and Vest (1979). The exposition that follows is synthesized primarily from these references, particularly the first three. In some cases, the words follow closely those of the the authors; their descriptions have become accepted as the classical standards. The references mentioned also carry extensive and useful bibliograpies.

The speckle effect

The invention of the laser created great anticipation among users of optics because it appeared to be the answer to a great many illumination problems. Here was a source that produced a beam of light that was intense, collimated, narrow, monochromatic, and coherent. Disappointment soon followed.