Our collaborative article with Prof. Haldun M. Ozaktas is now out at Applied Optics. In this article, we have unlocked the theoretical relationship between the Beam Propagation Method (BPM), which is famous approximation technique for simulating the propagation of light in slowly varying optical media, and linear canonical transformations.
We showed that 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 inhomogeneous media and potentially allow mathematical tools and algorithms associated with these transforms to be applied to the BPM.
Paper: https://opg.optica.org/ao/fulltext.cfm?uri=ao-61-34-10275&id=522121
Abstract:
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 inhomogeneous 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.