I. Introduction
The Special Affine Fourier Transformation (SAFT), which was introduced in [1] , is an integral transformation associated with a general inhomogeneous lossless linear mapping in phase-space that depends on six parameters independent of the phase-space coordinates. It maps the position‘ and the wave number into \begin{equation} \left(\begin{array}{l}x^{\prime } \\ k^{\prime } \end{array}\right) = \left(\begin{array}{ll}a & b \\ c & d \end{array} \right)\left(\begin{array}{l}x\\ k \end{array}\right) + \left(\begin{array}{l}p\\ q \end{array} \right), \tag{1} \end{equation}
with
\begin{equation}
ad-bc=1 . \tag{2}
\end{equation}
This transformation, which can model many general optical systems
[1], [5], maps any convex body into
another convex body and Eq.(2) guarantees that the area of the body
is preserved by the transformation. Such transformations form the inhomogeneous special linear group ISL