DDM theory

To obtain \(A\) and \(B\) when performing DDM some Fourier analysis is required. Starting from the definition of the image structure function,

(1)\[ \begin{equation} D(\vec{q}, t) = A(\vec{q})(1-f(\vec{q}, t) + B(\vec{q}), \end{equation} \]

which is calculated as

(2)\[ \begin{equation} D(\vec{q}, t) = \langle|\tilde{I}(\vec{q}, t + \tau) - \tilde{I}(\vec{q}, t)|^2\rangle_t. \end{equation} \]

Here, \(\tilde{I}(\vec{q}, t)\) is the Fourier transform of the image at time \(t\) and \(\tau\) is the lag time. By expanding equation (2) we can write this as

(3)\[ \begin{equation} D(\vec{q}, t) = 2\langle|\tilde{I}(\vec{q}, t)|^2\rangle_t - 2\langle\Re[\tilde{I}(\vec{q}, t+ \tau)\tilde{I}^\ast(\vec{q}, t)]\rangle_t. \end{equation} \]

To get to this point, there is a simplification made that \(\langle|\tilde{I}(\vec{q}, t+\tau)|^2\rangle_t\) is independent on \(\tau\), i.e. \(\langle|\tilde{I}(\vec{q}, t+\tau)|^2\rangle_t = \langle|\tilde{I}(\vec{q}, t)|^2\rangle_t\). By comparing terms dependent and independent of \(\tau\) in equations (3) and (1), one can write

(4)\[ \begin{equation} A(\vec{q}) + B(\vec{q}) = 2\langle|\tilde{I}(\vec{q}, t)|^2\rangle_t \end{equation} \]

and

(5)\[ \begin{equation} A(\vec{q})f(\vec{q}, t) = 2\langle\Re[\tilde{I}(\vec{q}, t+ \tau)\tilde{I}^\ast(\vec{q}, t)]\rangle_t. \end{equation} \]

At this point we can do one of two things. Firstly, we can assume that at very large \(q\), \(A(\vec{q}) = 0\). This is because \(A\) represents an amplitude term which depends on spatial intensity correlations. Therefore one can calculate

\[ \begin{equation} B(\vec{q}; q \to \infty) = 2\langle|\tilde{I}(\vec{q}; q \to \infty, t)|^2\rangle_t, \end{equation} \]

where \(q = |\vec{q}|\). It is then simple to calculate \(A(\vec{q})\) for all \(\vec{q}\) from equation (4).

Alternatively, we can use a priori knowledge of the Intermediate Scattering Function (ISF), \(f(\vec{q}, \tau)\) to say that \(f(\vec{q}, \tau \to 0)\). With this and equation (5) we can calculate \(A(\vec{q})\) as

\[ \begin{equation} A(\vec{q}) = \lim_{\tau\to 0}2\langle\Re[\tilde{I}(\vec{q}, t+ \tau)\tilde{I}^\ast(\vec{q}, t)]\rangle_t. \end{equation} \]