We show that for a wave ψ in the form of an Airy function the probability density ‖ψ‖2 propagates in free space without distortion and with constant acceleration. This ’’Airy packet’’ corresponds classically to a family of orbits represented by a parabola in phase space; under the classical motion this parabola translates rigidly, and the fact that no other curve has this property shows that the Airy packet is unique in propagating without change of form. The acceleration of the packet (which does not violate Ehrenfest’s theorem) is related to the curvature of the caustic (envelope) of the family of world lines in spacetime. When a spatially uniform force F (t) acts the Airy packet continues to preserve its integrity. We exhibit the solution of Schrödinger’s equation for general F (t) and discuss the motion for some special forms of F (t).

The evolution of a wave starting at z = 0 as exp(iαφ) (0 φ < 2π), i.e. with unit amplitude and a phase step 2πα on the positive x axis, is studied exactly and paraxially. For integer steps (α = n), the singularity at the origin r = 0 becomes for z > 0 a strength n optical vortex, whose neighbourhood is described in detail. Far from the axis, the wave is the sum of exp{i(αφ + kz)} and a diffracted wave from r = 0. The paraxial wave and the wave far from the vortex are incorporated into a uniform approximation that describes the wave with high accuracy, even well into the evanescent zone. For fractional α, no fractional-strength vortices can propagate; instead, the interference between an additional diffracted wave, from the phase step discontinuity, with exp{i(αφ + kz)} and the wave scattered from r = 0, generates a pattern of strength-1 vortex lines, whose total (signed) strength S α is the nearest integer to α. For small |α − n|, these lines are close to the z axis. As α passes n + 1/2, S α jumps by unity, so a vortex is born. The mechanism involves an infinite chain of alternating-strength vortices close to the positive x axis for α = n + 1/2, which annihilate in pairs differently when α > n + 1/2 and when α < n + 1/2. There is a partial analogy between α and the quantum flux in the Aharonov-Bohm effect.

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Scalar or vector light of wavelength 2/k strikes N small refracting and absorbing spherules, each of radius a, which have coagulated into a sparse random cluster with fractal dimension D (for smoke, D 1 78). It is assumed that ka<< 1 but the cluster size R = aNlID may be larger than the wavelength. Using a mean field theory it is shown that multiple scattering is negligible for all N if D <2, and becomes important when N-(ka) -D(D -2) if D> 2. Cross-sections are calculated as functions of N, D and the complex refractive index of the spherules. If D < 2 the scattering cross-section per spherule rises with N and saturates when kR >> 1, at a value exceeding that of an isolated spherule by a factor of order (ka) -D; if D> 2 the same quantity increases as N l ' -2/D/(ka) 2 . For D<2 the absorption cross-section is N times that of a solitary spherule. These results are very different from those for spherules coagulating into compact solid spheres (D = 3), and are important for the optics of sooty smoke, with Dr1-78, which is used as an example throughout. IntroductionSmoke is formed as small soot spherules [1] which stick together when they touch and thereby coagulate. Further aggregation leads to sparse random clusters with a fractal dimension of about 78 2, 3]. This fractality has not been included in studies of the optics of smoke. The omission is important for the scattering and extinction of radiation by smoke lofted into the atmosphere by large fires, such as those produced by multiple nuclear explosions [4][5][6][7]. Our purpose here is to present a theory of the optics of smoke in which the importance of the fractal structure is fully brought out. Implications for the nuclear winter predictions will be discussed in a separate publication.The central simplifying principle will be that the wavelength of the illuminating radiation, which is about 05 for the visible and 10 p in the infrared, greatly exceeds the spherule radius a, which is about 0005 u-0 0 5 p for sooty smoke. In other words, the observation scale is much bigger than the inner fractal scale. However, the size R of a cluster (that is, the outer fractal scale) increases with aggregation ( § 2) as the number of spherules in a cluster increases, from a (when N= 1) to values which may considerably exceed (when N is several thousand). We shall calculate the optical cross-sections as functions of N and show that the optics of clusters with R<>A can be very different. Moreover the differences depend strongly on whether the fractal dimension D is less than 2 or greater than 2. Downloaded by [Monash University Library] at 00:

T h e exact determinantal equation satisfied by the coherent nave vector in a statistically defined medium of spherical scatterers, which was obtained by Lloyd in 111 of this series of papers, is used to show two things. First, the determinant can be written as an infinite series similar to that obtained by ordinary resummation methods, except that it does not involve elements of the T matrix off the energy shell. Secondb, the first two terms of the series are specialized to the case of point scatterers which are uncorrelated except for an infinitesimal sphere of exclusion about each of them, and the results found to disagree with those of other calculations; this discrepancy is clarified. The formalism is generalized to deal with vector waves, and it is shown that, for electric dipole scattering, the simplest approximations to the functions appearing in the determinant lead to the Lorentz-Lorenz law.