Abstract
A powerful technique to solve differential equations - both ordinary and partial - with constant coefficients is the integral transform. The basic idea is to replace the unknown solution to the differential equation with a suitable integral of this function that contains an unknown parameter. If we can do this, then ordinary differential equations are reduced to algebraic equations, and partial differential equations are replaced by one of lower dimensionality.
The linear canonical transform is a four-parameter class of linear
integral transforms. These linear canonical transforms have been developed by many researchers under different names and different contexts. They are variously known as the generalized Fresnel transforms, ABCD transforms, Collins formula, generalized Huygen’s integrals, Moshinsky and Queesne integrals, and as a special case of the special affine Fourier transform. The fact that these transforms have been used in so many different contexts implies the universal importance of the transform. They provide a unified mathematical framework of the many tools we use in optics.
For example, the Fresnel transform studied for light propagation, the theories developed for quadrature phase filtering in signal processing, and theories developed for chirp pulse propagation in radar, optical communication, and for echo bat propagation all share the same linear canonical transform with the exception of one parameter. This is a major area of investigation in optical science, but, unfortunately, we do not have space to go deeply into this. The reader is advised to refer to the literature for further details.
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