We now show how CAM theory deals with the first of the semiclassical critical effects, forward diffractive scattering. The impenetrable sphere model is ideal for this purpose, because the only relevant physical processes that take place in it are reflection from the surface and forward diffractive scattering.

The difficulties are connected with the transition domain between the forward diffraction peak and the wide-angle geometrical reflection region, where the WKB approximation holds (Sec. 7.5). Within the forward peak, classical diffraction is dominant, but this domain of small diffraction angles is only sensitive to the bulk blocking effect of the scatterer (Sec. 2.2). To probe the dynamics of diffraction one must go to larger diffraction angles, contained within this difficult transition domain.

Fock's theory of diffraction is not powerful enough to bridge the gap, as will be seen in several examples. For this purpose, one needs a uniform asymptotic approximation. Such an approximation was developed by Berry (1969) for forward diffractive scattering by potentials with long-range tails, but near-forward scattering in this case arises from large impact parameters, a physical mechanism completely different from diffraction by the edge of a curved surface.

The crucial physical concept is that of the effective potential. It will be seen to lead to a reinterpretation of Fock's theory, clarifying several points that remained obscure in it (Sec. 7.3).

In Chapters 9 to 11, the discussion of Mie scattering was restricted to N > 1. We now consider a new type of semiclassical diffraction effect, not included in the classification given in Chapter 1, that takes place, for N < 1, near the critical angle for total reflection. It occurs, for example, in the scattering of light by air bubbles in water.

In the geometrical-optic limit, the new effect, near-critical scattering, does not correspond to a caustic singularity in the light intensity, which remains continuous: there is a limiting singularity, but only in the gradient of the intensity. We refer to it as a weak caustic.

The penetration of light incident beyond the critical angle into the optically rarer medium is the earliest example of tunneling in physics. It was discovered, amazingly enough, by Newton, who observed frustrated total reflection in his experiments on Newton's rings (see the above quotation).

According to Newton's ideas, a light ray would follow a parabolic path within the rarer medium before returning to the denser one. This would produce a lateral displacement of the reflected beam. Although our views about paths differ from Newton's, the displacement does exist: it is known as the Goos–Hänchen shift (Goos & Hänchen 1947). In near-critical scattering from a curved interface, it is modified by the dynamical effects of curvature.

The main theories that have been proposed to explain the observed features of the rainbow, from Descartes to the present, are reviewed here. They evolved in close parallel with theories about the nature of light. Surprisingly, as late as 1957, it was pointed out that no reliable quantitative theory of the meteorological rainbow existed at that time (apart from complicated numerical computer calculations).

Geometrical-optic theory

In the rainbow (for color pictures, see Greenler 1980), the primary bow is surrounded by the much fainter secondary bow, in which the colors appear in reverse order. The bows are just sets of directions in the sky, centered around the antisolar point, the direction opposite to that of the Sun: the higher the Sun's elevation, the lower the antisolar point below the horizon, and the smaller the portion of the rainbow arcs seen.

Each of the bows has a bright side and a shadow side, and the shadow sides face each other, so that the sky between the bows looks darker, as was described by Alexander of Aphrodisias around 200 AD (Alexander's dark band).

By means of an experiment performed at the remarkably early date of 1304, it was demonstrated by Theodoric of Freiberg that the rainbow arises from the scattering of sunlight by individual water droplets in the atmosphere.

In Chapter 4, we discussed the optics necessary for system designing to record and reconstruct an in-line Fraunhofer hologram. A single microobject model and the simplest mode of recording and reconstruction were considered. The technique often finds limitations in practical situations. For example, if the density of the microobjects in the test-section is high, no reference beam might be present in the transmitted light. Thus, there is a practical limit on the density for the successful recording and hence reconstruction. In some other situations, the ideal position of the recording medium may not be practical. This, for example, could be inside a combustion chamber. The scene in that situation can be relayed to a convenient location using lenses. A number of these practical limitations and possible ways to circumvent them are considered in this chapter.

Techniques to enhance the image quality such as special recording and/or processing of the hologram, spatial frequency filtering methods, etc. are also described.

Density of microobjects

In Chapter 3 we saw that an in-line hologram is formed by the interference between the directly transmitted light and the diffraction pattern produced by the microobject. For a single microobject, there is no problem because the reference beam is present almost everywhere. When the number of microobjects increases in a given cross-sectional area, the effective presence of the reference beam will reduce and hence the hologram recording process will be affected. In fact, this is the simple way to describe the situation.

The standard semiclassical approximations in scattering theory break down in four well-known situations, that correspond to some of the most interesting scattering effects. In order to treat them, one must be able to handle diffraction, a notoriously difficult subject involving the dynamical aspects of wave propagation.

The basic theme of this book is how to deal with these critical effects. Fortunately for the theorist, there exists an exactly soluble model that exhibits all of them, the scattering of light by a homogeneous spherical particle. Some of the most beautiful phenomena in nature, coronae, rainbows and glories, are visual manifestations of the critical effects contained within this model. The model is not only soluble: it describes with extremely high accuracy the actual behavior of small liquid droplets.

The formally exact solution of the light scattering problem, obtained by Mie at the beginning of this century (and even earlier by Lorenz), is in the form of an infinite series, the well-known partial-wave series; analogous problems for elastic waves had been solved in this form several decades before. During this period spanning over a century, many important contributions to mathematical physics, particularly in connection with special functions and asymptotic approximation methods, originated in the treatment of these problems.

Unfortunately, the series converges very slowly under semiclassical conditions. In view of the manifold practical applications, substantial effort has been devoted to developing fast computer programs for the numerical summation of the series solution.

We now begin to apply the CAM method to Mie scattering. The penetration of the field and the resulting interactions within the scatterer yield a rich structure, where all semiclassical diffraction effects are found. One consequence is that a preliminary transformation of the scattering amplitude is required before applying the Poisson representation to obtain fast convergence.

This transformation, first employed by Debye (1908) in scattering by a circular cylinder, is analogous to the multiple internal reflection treatment of the Fabry–Perot interferometer (Born & Wolf 1959). In a geometrical-optic approximation, it corresponds to the ray-tracing procedure illustrated in fig. 2.1: the scattering amplitude is decomposed into an infinite series of terms representing the effects of successive internal reflections.

In the present chapter, we discuss the results obtained by the CAM method for the first two terms of this series, that represent the effects of direct reflection from the surface and direct transmission through the sphere, employing transitional approximations (Nussenzveig 1969a, Khare 1975). One new feature found is the penetration of diffracted rays within the scatterer, taking ‘shortcuts’ through it before reemerging as surface waves.

The effective potential

The Mie scattering amplitudes were given in Sec. 5.1. Unless otherwise stated, we assume that the sphere is perfectly transparent, so that the relative refractive index N is real. This is the hardest case to treat: absorption damps out contributions from all but the shortest paths through the sphere and tends to greatly improve convergence; the CAM treatment remains valid for complex N.

In this and in the following chapter, we present a very sketchy and incomplete survey of selected applications and illustrations of semiclassical diffraction effects in various branches of physics. The purpose is to furnish a sample of the variety of areas in which manifestations of similar phenomena are present, emphasizing recent developments, as well as to provide some guidance to the literature.

Recent applications of Mie scattering

The reader is referred to previous monographs (van de Hulst 1957, Kerker 1969, McCartney 1976, Bohren & Huffman 1983) for surveys of traditional applications of light scattering by spherical particles. Detailed discussions of coronae, rainbows and glories in meteorological optics are given in Linke & Möller 1961. We review a couple of more recent applications.

Glare spots

When a suspended liquid droplet is illuminated by a parallel light beam and viewed from a relatively close distance (such that it subtends an angle of a few degrees at the observation point), a number of glare spots are seen on its surface; as the observer moves around the droplet, glare spots merge and become colored when rainbow angles are crossed (Tricker 1970, Walker 1976, 1977, 1978). Glare spots are clearly visible in the color pictures of droplets reproduced in Qian et al (1986).

The observer's distance, though close, is still far enough that the field is given by the far-zone expressions (5.1).

Once a hologram of an event is successfully recorded, the reconstruction at proper net magnification will yield images of the microobjects distinct in the background. As seen in Chapter 3, the image represents the object cross-section encountered at the time of recording. Generally knowledge of net transverse magnification is required for size analysis. Information about the volume magnification is also often needed for determining the density of microobjects.

The net transverse magnification is generally several hundred (see Section 4.9). This results in a limited field of view on a fixed cross-section observation screen. To cover a considerable subject or image volume, either a detector such as the television camera or the hologram on a translation stage is moved for frame by frame analysis. For the general case of a spherical recording and/or reconstruction beam, the magnifications are not straightforward.

The transverse magnification due to the camera-lens system is fixed for a given lens–screen (or vidicon) separation. This magnification can be easily determined by calibration using a resolution chart or a transparent ruler scale. However, the holographic magnification due to the spherical beams is space variant. These magnifications are described in detail in this chapter.

The analysis is generally valid for in-line as well as off-axis holography. We start with the reconstructed image position known in terms of the object, reference and reconstruction source points, and wavelengths of light sources. The needed magnifications are then discussed for modes suited to users of the technique.

The paraxial approximation or first-order optics considered so far in the analysis for determining the image coordinates and magnification (Chapter 6) is sufficient for many applications. As the minimum required effective hologram aperture increases for smaller objects, the aberrations must be considered for accurate imaging and analysis. This chapter is devoted to the general background development for third-order aberrations dealing with particle field holography. The in-line method is considered in more detail due to its widespread use. The reference source, the object and the hologram center points form a straight line in the in-line method, leading to interesting aspects of aberrations and their control. Cylindrical test-sections with curved windows are also encountered in practical far-field holography. Cylindrical ampules, water tunnels, etc. are, in effect, cylindrical lenses resulting in unwanted astigmatism. Ways to neutralize the cylindrical lens are therefore also discussed in the chapter.

Primary aberrations

For determining the coordinates of the image point (Chapter 2), the object, reference and reconstruction beams were assumed to be spherical over the hologram aperture. This approximation neglected higher order terms in the phase expressions of the wavefronts. Since particle field holography deals with very small objects, the aberration allowances are also small. In the presence of aberrations, diffraction-limited resolution may not be obtained.

From the discussion of Chapter 2, the ideal image point will form a spherical phase term over the hologram aperture with its center at the image point.