We now deal with the last of the critical effects listed in Chapter 1: orbiting, which, as was mentioned in Sec. 1.3, is often associated with the presence of resonances. This is also true here.

The main feature of Mie scattering that we have not treated so far, the existence of sharp ripple fluctuations for ReN > 1, is due to resonances. These resonances, in the idealized model of a perfectly transparent sphere, grow arbitrarily sharp with increasing size parameter; in real life, they can still be extremely narrow, leading to a variety of interesting applications that will be described in Chapter 15.

The origin and physical interpretation of the resonances are greatly clarified by employing the effective potential picture. The resonances appear as ‘quasibound states of light’, associated with orbiting-like paths of light rays around the scatterer. Tunneling through the centrifugal barrier plays an essential role in determining the resonance widths and in explaining the ‘sensitivity to initial conditions’ that characterizes the ripple fluctuations.

CAM theory describes the resonances in terms of families of Regge trajectories, a concept that has found important applications in high-energy physics (Collins 1977). It allows one to determine the resonance positions and widths with high accuracy.

The resonance contributions correspond to residues at the Regge poles, following their trajectories. By adding them to the background that results from the other effects already treated by CAM methods, one can finally obtain complete fits of the Mie cross sections, including the ripple.

When absorption within the scatterer is taken into account, the incident beam is extinguished both by scattering and by absorption, and one associates a specific cross section with each of these processes. In applications to radiative transfer, it is important to consider the balance not only of energy but also of momentum: this is related with the radiation pressure cross section.

All of these cross sections, for Mie scattering, are plagued by ripple fluctuations similar to those discussed in Sec. 5.2, complicating considerably their numerical evaluation. These fluctuations are irrelevant for most applications, because they are quenched by size dispersion, and one usually deals with a polydisperse population of scatterers.

Thus, one would like to obtain average cross sections, where the average is taken over a size interval just large enough to eliminate the ripple fluctuations, while still allowing slower-scale variations to be resolved. A similar problem occurs in the application of the nuclear optical model to the theory of nuclear reactions (see the above quotation), and one might describe the procedure employed as an ‘optical application of the optical model’.

In special circumstances, one may observe, instead of the irregular ripple fluctuations, smooth oscillations of the cross sections about their averages, arising from forward optical glory effects, predicted and explained by complex angular momentum theory.

The general procedure of the analysis is to bring the reconstructed image in focus at the final observation plane. This mode of analysis is very common. However, particularly in the in-line method, there are proposals to perform the analysis at other planes. These are: the recording plane (hologram) itself, the misfocused image plane, and certain transform (Fourier) planes. These approaches have advantages in particular situations. In this chapter, these techniques of non-image plane analysis are discussed. Non-image plane analysis techniques have applications in velocimetry also. Those methods are dicussed in Chapter 10.

Analysis of the far-field pattern

The pattern at the recording plane of the Fraunhofer hologram itself contains information about the object and its location. The possibility of extracting the desired information from the pattern is as old as Fraunhofer holography itself. Thompson and coworkers proposed utilizing the pattern itself for the analysis at the early stages of the development of far-field holography. There are two approaches. One is to study the entire complex pattern to determine the size and the distance of the microobject. This is called the coherent background method due to the presence of the reference beam. The other method, called the spatial filtering method, yields the broad diffraction pattern for further analysis.

Coherent background method

The principle of the technique can be described from a typical Fraunhofer diffraction pattern. The schematic diagram is shown in Figure 9.1.

We start by reviewing the history of glory observations and their consequences, which is almost as remarkable as the phenomenon itself. We describe various proposed explanations of the effect: until recently, the best one available was based on a conjecture by van de Hulst, related with the geometrical theory of diffraction, which is also briefly discussed.

Observations

The earliest recorded observations of the glory were made between 1737 and 1739 (cf. Lynch & Futterman 1991), during a French geodetic expedition to Peru, undertaken to settle the dispute between Newton and Cassini on the figure of the Earth (occasioning Voltaire's famous quip: ‘Vous avez confirmé dans des lieux pleins d'ennui, Ce que Newton connut sans sortir de chez lui’). Early one morning, a few members of the expedition, including Bouguer and a Spanish captain, Antonio de Ulloa (who introduced platinum into Europe following this expedition), stood on top of Mount Pambamarca, in the Peruvian Andes.

It was with some considerable pleasure that I accepted Chandra Vikram's invitation to prepare a foreword for his book on particle field holography. He has made major contributions to this field – both to its science and its technology. This book is a fine summary of the current state-of-the-art and goes hand-in-hand with the recently published SPIE Milestone Series volume on Selected Papers on Holographic Particle Diagnostics (MS21).

Certainly it is always flattering to be asked to write a foreword to a volume such as this. In this particular case, however, I happen to have a special relationship to the subject matter of particle field holography. I had the good fortune to be one of the co-founders of this particular area of endeavor and have continued to be involved in it from time to time with some modest contributions of my own, but more importantly with some significant work jointly with professional colleagues and with first-rate graduate students as part of their individual Ph.D. theses.

It is hard to believe that this particular application of holography started nearly 30 years ago and in a rather interesting way. As I chronicled in a recent review paper.

Two or more recordings of moving microobjects can be recorded to the same recording medium. The different sets of images provide the displacement and other changes (such as shape and change in size of burning coal particles). Non-image plane analysis techniques (see Chapter 9) have also been applied for velocimetry applications. Different reference or object beam directions for storing different frames provide another way of multiplexing. The techniques of high speed photography have also been extended to holographic applications. In this chapter we will discuss the methods applicable to particle field holography.

Multiple-exposure holography

The object–image point relationships (see Chapter 6) are known. The concepts of high speed photography can be used there to extract the velocity and other time variant changes with the object field. The advantage of using holography is that microobjects in a volume with changes in three dimensions can be studied. As reported by Boettner and Thompson, a microobject can be allowed to move during the exposure to obtain a streak yielding the path of the movement. Trolinger and coworkers introduced the use of multiple exposures to study the dynamics of the particle field. Multiple exposures can be provided, for example by several lasers with synchronized fires, or more conveniently by active or passive Q-switches. Trolinger, Belz and Farmer reported several double exposure holograms and reconstructions demonstrating the usefulness of the technique in velocity and other transient phenomena studies. A double-exposure hologram of the moving particle field reconstructs two sets of images.

The study of very small objects in a dynamic volume is of significant importance in modern science and technology. Particles, bubbles, aerosols, droplets, etc. play key roles in processes dealing with nozzles, jets, combustion, turbines, rocket engines, cavitation, fog, raindrops, pollution, and so on. The quantities to be studied are size, shape, velocity, phase, etc. Holography provides an excellent imaging tool with aperture-limited resolution in a volume rather than just in a plane. The purpose of this book is to describe this subject matter under one cover for the first time.

After a brief historical background in Chapter 1, Chapter 2 introduces the fundamental expressions and notations of holography to be used throughout the book. Chapter 3 covers general theory and mathematical formulations of in-line far-field holography. The recording and reconstruction processes are discussed in detail and the relationships obtained are used throughout the book. Chapters 4 and 5 deal with a large number of the design and practical problems often encountered. Aspects like film size and resolution requirements, source coherence requirements, allowable object velocity, hologram recording and processing techniques, image enhancement, ultimate resolution limits and uncertainties are detailed in these chapters. Chapter 6 provides detailed analysis to relate object and image spaces. The aspects covered are location and magnifications for quantitative analysis of the image. Chapter 7 covers third-order aberrations dealing with holography of small objects. Limits due to these aberrations on the ultimate resolution and some ways to control the aberrations are described.

There are a large number of situations in modern science and technology requiring the study of very small objects in a volume. These small objects, for example, could be fog, sprays, dust particles, burning coal particles, cavitation and other bubbles in water, fuel droplets in a combustion chamber, etc. In practical situations these objects are small and micrometer order resolution is often required. This resolution itself is not a problem with conventional microscopic techniques. However, a serious problem arises due to the need to study a dynamic volume and not just a plane. An imaging system that can resolve a diameter d has a depth of field of only about d2/λ, where λ is the wavelength of the light used. For d = 10 μm, λ = 0.5 μm, the depth of field is only 0.2 mm! Clearly this is not satisfactory.

Light scattering and diffraction methods depend on models. These models are often based on assumptions of the object shape, pre-knowledge of the refractive index and other physical parameters. These assumptions are good enough when rapid and mean particle size distribution rather than exact shape and other parameters are needed. An example is size distribution measurements in the ceramic industry. These methods being real-time in nature, are very valuable in particular situations. Some of these methods are described in Chapter 9.

These non-imaging methods are not adequate in a large number of situations of practical interest.

The Mie series solution for the scattering of an electromagnetic plane wave by a homogeneous sphere is introduced. Though ‘exact’, ‘a mathematical difficulty develops which quite generally is a drawback of this “method of series development”: for fairly large particles … the series converge so slowly that they become practically useless’ (Sommerfeld 1954). What is still worse, numerical studies reveal the occurrence in Mie cross sections of very rapid and complicated fluctuations, known as the ‘ripple’, that are extremely sensitive to small changes in the input parameters. They are related to orbiting and resonances, which may also be detected through their contribution to nonlinear optical effects, such as lasing, that are observed in liquid droplets.

The Mie solution

We consider a monochromatic linearly polarized plane electromagnetic wave with wave number k incident on a homogeneous sphere of radius a and a complex refractive index (relative to the surrounding medium) N = n + iκ, where n is the real refractive index and κ is the extinction coefficient (Jackson 1975). The time factor exp(–ωt) where ω = ck is the circular frequency, is omitted.

Taking the origin at the center of the sphere, the z axis along the direction of propagation of the incident wave and the x axis along its direction of polarization, the incident electric field is E0 = exp(ikz) ◯. In spherical coordinates, the components of the scattered electric field at large distances are of the form (Bohren & Huffman 1983)

CAM theory provided for the first time a detailed physical explanation of the meteorological glory. It confirmed van de Hulst's conjecture (Sec. 4.3) about the importance of surface waves of the type illustrated in fig. 4.3, but it showed that higher-order Debye contributions, both from surface waves and from the shadow of higher-order rainbows, are very important. What is very remarkable is that all these contributions arise from complex critical points, so that the glory is a macroscopic tunneling effect. Several other physical effects must be taken into account, rendering the glory one of the most complicated of all known scattering phenomena.

The meteorological glory is observed in backscattering. CAM theory predicts the existence as well of a forward optical glory, that will be discussed in Chapter 13.

Observational and numerical glory features

Observations of natural glories, together with laboratory observations, have shown some conspicuous differences between glories (sometimes called anti-coronae) and coronae (van de Hulst 1957).

1. (i) Variability. The appearance of the glory rings varies considerably from one observation to another, and sometimes even in the course of a single observation. Indeed, as will be illustrated below, the near-backward intensities are rapidly varying functions of β, θ and N.

2. (ii) Angular distribution. The angular distribution falls off away from the central region much more slowly than in the coronae, so that the outer rings are more pronounced: the observation of as many as five sets of rings has been reported.

3. […]

Introduction

The theoretical studies of long-distance soliton transmission in optical fibres with periodic Raman gain were outlined in Chapter 2 ‘Solitons in Optical Fibres: An Experimental Account’ by L. F. Mollenauer. From this work, it was anticipated that the overall information rates made possible by the all-optical nature of the system were potentially orders of magnitude greater than in conventional communication systems. Rate–length products were predicted to be as high as approximately 30000 GHz · km for a single wavelength channel. In the following sections we will describe the first set of experiments designed to explore the fundamentals of such long-distance repeaterless soliton transmissions.

Experimental investigation of long-distance soliton transmission

In the following experiments, a train of soliton pulses was injected into a closed loop of fibre, the length of which corresponded to one amplification period in a real communication system. The pulses propagated around the loop many times until the net required fibre pathlength had been reached (Mollenauer and Smith, 1988). The experimental configuration is shown schematically in Figure 3.1. A 41.7 km length (L) of low-loss (0.22 dB/km at 1.6 μm) standard telecommunications single-mode fibre was closed on itself with a wavelength selective directional coupler (an all-fibre Mach–Zehnder (FMZ) interferometer). The interferometer allowed efficient coupling of the bidirectional pump waves at approximately 1.5 μm (provided by the 3 dB coupler at the pump input) into the loop. At the same time, the signals at approximately 1.6 μm (50 ps sech2 intensity profile pulses at a 100 MHz repetition rate from a synchronously mode-locked NaCl:OH– colour centre laser) were efficiently recirculated (<5% coupled out).

Introduction

In a communication system, it is desirable to launch the pulses close to each other so as to increase the information-carrying capacity of the fibre. But the overlap of the closely spaced solitons can lead to mutual interactions and therefore to serious performance degradation of the soliton transmission system. This has been pointed out independently by three groups (Chu and Desem, 1983a; Blow and Doran, 1983; Gordon, 1983).

Karpman and Solov'ev (1981) first considered the two-soliton interaction in their study of the non-linear Schrödinger equation (NLS) by means of single-soliton perturbation theory. Although they did not have optical fibre transmission in mind their results are applicable to fibres since the soliton propagation in optical fibres are described by the same equation. However, their method is restricted to large soliton separation only.

Our numerical investigations (Chu and Desem, 1983a) show that soliton interaction can lead to a significant reduction in the transmission rate by as much as ten times. At about the same time, Blow and Doran (1983) showed that the inclusion of fibre loss also leads to dramatic increase of soliton interactions. Gordon (1983) derived the exact solution of two counter-propagating solitons (of nearly equal amplitudes and velocities) and analysed the interaction by obtaining the approximate equations of motion corroborating the results of Karpman and Solov'ev (1981).

A considerable amount of research effort has been spent on the reduction of soliton interactions.

Introduction

Although soliton phenomena arise in many distinct areas of physics, the single-mode optical fiber has been found to be an especially convenient medium for their study. As described in the previous chapters of this book, soliton propagation of bright optical pulses has been verified in a number of elegant experiments performed in the negative group velocity dispersion (GVD) region of the spectrum; most recently, transmission of 55-ps optical pulses through 4000 km of fiber was achieved, by use of a combination of non-linear soliton propagation to avoid pulse spreading and Raman amplification to avoid losses (Mollenauer and Smith, 1988). For positive dispersion (λ < 1.3 μm in standard single-mode fibers), bright pulses cannot propagate as solitons and the interaction of the non-linear index with GVD leads to spectral and temporal broadening of the propagating pulses. These effects form the basis for the fiber-and-grating pulse compressor (Nakatsuka et al., 1981; Grischkowsky and Balant, 1982; Tomlinson et al., 1984), which was utilised to produce the shortest optical pulses (6 fs) ever reported (Fork et al., 1987). For both signs of GVD, the experimental results are in quantitative agreement with the predictions of the non-linear Schrödinger equation (NLS).

Although bright solitons are allowed only for negative GVD, the NLS admits other soliton solutions for positive GVD (Hasegawa and Tappert, 1973; Zakharov and Shabat, 1973). These solutions are ‘dark pulse solitons’, consisting of a rapid dip in the intensity of a broad pulse of a c.w. background.