The beam propagation method (BPM) can be viewed as a chain of alternating convolutions and multiplications, as filtering operations alternately in the space and frequency domains or as multiplication operations sandwiched between linear canonical or fractional Fourier transforms. These structures provide alternative models of in homogeneous media and potentially allow mathematical tools and algorithms associated with these transforms to be applied to the BPM. As an example, in the case where quadratic approximation is possible, it is shown that the BPM can be represented as a single LCT system, leading to significantly faster computation of the output field.