Optics of Fractal Clusters Such as Smoke

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 = aN 1/D 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) m D/(D m 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 kRd 1, at a value exceeding that of an isolated spherule by a factor of order (ka) m D; if D 2 the same quantity increases as N 1 m 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 D , 1·78, which is used as an example throughout.