Linear canonical transformation

Summary

A linear canonical transformation is an integral transform of functions on $\R$ attached to a real $2 \times 2$ matrix of determinant $1$. It includes the Fourier transform, the fractional Fourier transform, the Fresnel transform, scalings and multiplication by chirps as special cases. Composing two such transforms gives, up to a sign, the transform of the product matrix. So they form a projective unitary representation of $SL(2, \R)$, which is a genuine representation of the metaplectic group. In optics it describes the propagation of light through lenses and free space in the paraxial approximation.