Consider two identical dipoles (Fig. 3.5) illuminated by a monochromatic (scalar) plane wave, or by a source with a lateral coherence length much greater than the separation between them, with wavenumber k = 2n/X. Assume that they are excited mostly by the incident wave, mutual excitation being negligible. Because the two dipoles are excited by the same incident wave, the waves scattered by them bear a fixed phase difference that depends on the separation between them, the wavelength of the illumination, and the direction of observation. Denote by e; the direction of the incident wave, by es1 and es2 the directions of the scattered waves (directions toward the point of observation O), and by r12 the position of dipole 2 relative to dipole 1. Although the scattered waves are more or less spherical waves, we take the point of observation to be sufficiently far away that they can be considered to be locally plane waves. We consider scalar waves even though light is a vector wave because we are interested only in the phase difference. If the complex amplitude of the incident wave at 1 is a, at 2 it is a exp(ike; • r12). Because the amplitude of each scattered wave is proportional to the

Figure 3.5: At O the total scattered field is the superposition of fields scattered by the two dipoles a fixed distance r12 apart. The phase difference between these two fields depends on the angle between the incident and scattered directions and the distance between the dipoles relative to the wavelength of the illumination.

Figure 3.5: At O the total scattered field is the superposition of fields scattered by the two dipoles a fixed distance r12 apart. The phase difference between these two fields depends on the angle between the incident and scattered directions and the distance between the dipoles relative to the wavelength of the illumination.

amplitude of the wave that excites it, the two scattered waves at O are

= aexp(ikesi ■ ri), = aexp(ikei ■ r12 + ikes2 ■ ri - ikes2 ■ r12), (3.92)

Any factors common to the two waves are omitted. As the distance to the observation point O increases indefinitely, es2 ^ es1 = es, and the phase difference between the two waves approaches

A(p = k(ei - es) • ri2 = ^-(e; - es) • ri2. (3.94)

From this simple equation flows an amazing amount of physical understanding.

The forward direction is the direction of the incident wave. According to Eq. (3.94), scattering in the forward direction (es = e;) is always in phase (A^> = 0) regardless of the separation of the two dipoles and the wavelength. This is the only scattering direction for which this is true, and hence the forward direction is singular, a point to which we return.

Any particle is a coherent array of N dipoles. Again, we assume that they are excited mostly by the incident wave. If the linear extent of this array is small compared with the wavelength, all the separate scattered waves are approximately in phase for all directions [see Eq. (3.42) with A^> = 0 and a1 = a2], and hence the power scattered in any direction by N scatterers is N2 times the power scattered by one. If the total volume of the particle is v and there are n dipoles per unit volume, the total number of dipoles is proportional to v. We therefore predict that the total power scattered by a particle small compared with the wavelength is proportional to v2. As we show in Section 3.5, this turns out to be correct.

Interactions between dipoles are not negligible unless separated by sufficiently large distances. Although this condition is not satisfied by a small particle, scattering still increases as the square of its volume even when interactions are accounted for (by interactions is meant that the wave from each dipole appreciably excites its neighbors). The reason for this is that if a particle is sufficiently small, appreciable phase differences are not possible even with interactions. Hence the array of dipoles (particle) is approximately coherent and in-phase even though the dipoles excite each other.

Let us continue in this vein and see what general results we can obtain by physical reasoning based on Eq. (3.94). As the overall size of the particle increases, the phase differences between the waves scattered by its constituent dipoles also increase. When the particle is sufficiently small, all these waves are approximately in phase (constructive interference) for all scattering directions. But as the particle size increases, phase differences of order n or greater become possible for some directions. In particular, the phase difference between two or more dipolar waves can be such that in a particular direction they interfere destructively. Thus we conclude that although scattering by a particle increases as the square of its volume when the particle is small compared with the wavelength, this rapid increase cannot be sustained indefinitely. As particle size increases, scattering increases, but more slowly than volume squared. And again, this expectation is supported by detailed calculations (see Sec. 3.5.1). We can understand them, at least qualitatively, by simple phase difference arguments.

We show in Section 3.2 that scattering by a dipole (subject to restrictions) is approximately inversely proportional to the fourth power of the wavelength of the incident illumination. But this does not necessarily imply that scattering by a coherent array of such dipoles (a particle) also follows this wavelength dependence. As evidenced by Eq. (3.94), an additional wavelength dependence creeps in by way of the phase difference. Indeed, for sufficiently large particles, the net effect of the wavelength dependence of scattering by the individual dipoles nearly cancels the wavelength dependence originating from phase differences. In the following section we give a simple example to elaborate on this point.

Now let us turn to the directional dependence of scattering. Suppose that our two dipoles individually scatter the same in all directions. This does not then imply that scattering by the two together is the same in all directions. According to Eq. (3.94) the phase difference between the two scattered waves depends on the scattering direction. We can make this clearer by considering the special example of two dipoles on a line parallel to the incident wave (ri2 = re;), in which instance the phase shift (again, interactions are neglected) is

where d is the scattering angle (angle between incident and scattered waves). The power scattered in any direction by the two dipoles is determined by cos Ay. The quantity 1 - cos d lies between 0 (forward direction) and 2 (backward direction), and hence the phase difference lies between 0 (forward) and 4nr/A (backward). When the (backward) phase difference is an odd multiple of n, interference is destructive; when the phase difference is an even multiple of n, interference is constructive; and, of course, everything between these two extremes is possible. As r/A increases so does the number of oscillations in the scattering diagram (scattering as a function of angle).

As a particle increases in size so does the average distance between its elements. We therefore expect, on the basis of the behavior of two dipoles, the scattering diagram for a particle to exhibit more maxima (minima) the greater its size relative to the wavelength. This expectation is borne out by detailed calculations (Sec. 3.5.1) as well as by measurements.

Because scattering is in-phase in the forward direction regardless of the wavelength and the separation between dipoles, we expect scattering in this direction to increase more rapidly with size than scattering in any other direction. Again, this expectation is borne out by calculations (Sec. 3.5.1) and measurements. A general result is that the larger the particle, the more that scattering by it is peaked in the forward direction.

The rate of change of the phase difference with respect to r ranges from 0 in the forward direction to 4n/A in the backward direction. The implication of this is that if we have a fixed number of dipoles (fixed particle volume) and we move them around (change the particle shape), scattering is least affected in the forward direction and most affected in the backward direction. Again, this is supported by calculations and measurements.

Much of the basic physics of scattering by particles is embodied in the simple phase difference given in Eq. (3.94). The rest is details, some of which we give in Section 3.5, but first we discuss scattering by air and by water.

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