Light scattering in the form of laser diffractometry is a valuable tool in physics, biology, material science and aerosol science. Its utility is associated with its ability to record light scattering from microparticles in their natural state in a non-destructive manner. For example, the technique can be applied to biological cells and droplets in suspension without affecting their viability.
The information available, in principle, from light scattering includes particle shape, size and refractive index. These three factors alone however permit such a large variety of scattering patterns that the analysis of the general particle may be considered to be intractable. As a consequence, analysis is often restricted to spherical particles. Even within this reduced set, only two classes of particles have been extensively studied.
These are the isotropic dielectric homogeneous sphere and the coated sphere. The optimum conditions for the analysis of the particles are obtained when nearly complete quasicontinuous scattering patterns are available over in a fixed plane of detection. A standard experimental arrangement is used to collect scattered light.
Studies involving the Mie light scattering equations have shown they are more representative of the problem when written in the form of Discrete Gegenbauer series. In the process of deriving these series the angular part is factored out leaving functions that depend on the radial part only. This Discrete Gegenbauer Analysis can therefore be regarded as a partial inversion of the light scattering data. All Gegenbauer spectra have relatively simple envelope structures and terminate at relatively small orders. This order is closely related to the outer size parameter whereas the structure is closely related to the refractive index or internal size parameter. These two features has allowed us to develop fast and accurate methods for computing the inner and outer size parameters.
A number of fast and accurate computational procedures have been developed for calculating related light scattering functions. These are continuously being enhanced with particular attention being given to those that require the largest computational time-slice. The four most computationally demanding algorithms are those that (a) compute integer order Bessel functions of real and complex argument (b) numerically integrate the angular light scattering data (c) compute Gegenbauer Functions and (d) compute the Decomposition Factors for generating the Gegenbauer Irradiance spectra.
Methods used to speed up the computational process usually involve introducing a greater number of processors and/or implementing approximate methods for some of the more time consuming algorithms. A more fundamental approach has been studied by us in which the Mie scattering equations are rewritten in a form that is more appropriate for analysis and machine computation. We express the Mie multipole coefficients in terms of amplitude and phase functions, where the phase function is expressed in absolute form. This assists the process of analysis and allows the computation of light scattering amplitude and irradiance functions to be performed faster and to machine accuracy. Faster algorithms are made possible as we have removed the need to compute directly the relatively large number of Bessel functions.
One unresolved question in the field of light scattering is whether the irradiance profile of a particular scattering object is unique, alternatively stated; can two appropriately selected scattering objects of different sizes and/or refractive indices produce identical light scattering irradiance profiles? If they can then the irradiance profiles are not unique. We are able to compute the irradiance profiles of particles using the Mie light scattering equations but there is currently no analytical method of ascertaining the particle parameters from the irradiance profile. The latter is a so called inverse light scattering problem.
Inversions have been carried out on homogeneous and layered spheres by the detailed matching of theoretical Irradiance functions to the experimental scattering patterns [1-10]. Although the procedure is plausible it has yet to be justified as no proof presently exists that unique values of and will be obtained for a particular scattering pattern. Moreover, to prove uniqueness using this method would require that a distinctive best fit solution is found after an exhaustive search over all values of α and β. This is clearly impractical.
Our treatment follows the formalism of elementary particle scattering adopted in nuclear physics of analysing the scattering function in the far field. This illustrates the important role of angular orthogonal functions in the theory and leads to a reformulation of Mie theory. As a consequence, it has been shown how a complete scattering pattern can be transformed to give a new plot from which angular functions have been eliminated. The new presentation takes the form of a discrete Gegenbauer spectrum of the scattering data.