On the Extraction of Polydispersity Information in Photon Correlation Spectroscopy

Abstract

It is well known that the problem of recovery of polydispersity information in photon correlation spectroscopy is preceded by the necessity to invert a Laplace transform in which the data is necessarily sampled, of finite extent and corrupted by noise [1]. There is a large literature on the numerical inversion of the Laplace transform but a new approach was initiated in 1978 by McWHIRTER and PIKE [2] who discovered the eigenvalues and eigenfunctions of the transformation and were thus able to construct an “information theoretic” inversion procedure based on the well-known Nyquist sampling ideas of standard information theory. This approach has been pursued and applied to the polydispersity problem by OSTROWSKY et al. [3]. In this application the polydisperse distribution of diffusion constants p(Γ) is reconstructed at a set of sample points

$$\begin{array}{*{20}{c}} {\frac{{{{n}^{\pi }}}}{{{{\omega }_{o}}}}} \\ {{{\Gamma }_{n}} = e = \delta _{o}^{n}} \\ \end{array}$$

(1)

, where ω₀ is a “frequency cut-off” which limits the sampling rate, as does the Nyquist criterion in the analogous sampling theory case, and δ₀ is a “resolution ratio”. The value of ω₀, unlike the Nyquist theory, was shown in [2] to be a function of the noise on the data. The method, based on a simple linear least-squares procedure, and discussed with examples in [3,4] is known as the “exponential sampling” method.