A family of integral transforms associated with a two-by-two real matrix of determinant one. For , one optical normalization has kernel proportional to . It includes free-space Fresnel diffraction, quadratic phase multiplication and the Fourier transform. The optical amplitude normalization in one transverse dimension is with a consistent phase branch.
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A linear canonical transformation (LCT) is a specific type of mathematical transformation used in various fields, including optics, quantum mechanics, and signal processing, to change the representation of a system while preserving certain properties. In general, LCTs are employed to map one set of variables to another in such a way that the structure of the system remains intact.