Fractionalization of cyclic transformations in optics

Tatiana Alieva, Martin J. Bastiaans, and Maria Luisa Calvo

The paper starts with some preparatory work on three types of transformations often used in information processing: a convolution, a matrix-vector multiplication, and a general linear integral transformation. In particular we represent these transformations in a diagonalized form. We then focus on cyclic transformations, and consider as examples the Hilbert transfromation, and the ordinary Fourier transformation and its direct relatives: the cosine, sine, and Hartley transformations.

Ways to fractionalize transformations that can be diagonalized, are described in the main part of the paper. As an example of a convolution operator, we consider the fractional Hilbert transformation. The canonical fractional Fourier transformer, as it is usually applied in optics, is the main example of the fractionalization procedure of a matrix-vector multiplication with - in this case - the ray transformation matrix of such a transformer. With the fractional Fourier transformation properly defined, we can then immediately consider its relatives: the canonical fractional cosine, sine, Hartley, and Hankel transformations. More general ray transformation matrices will then be introduced, in particular ortho-symplectic matrices. It is shown that all first-order optical systems whose ray transformation matrices can be diagonalized, are similar (in the sense of matrix similarity) to the separable fractional Fourier transformer. This, once again, stresses the important role of the fractional Fourier transformation. A relatively short treatment of general integral transformations, in particular transformations with a complete set of eigenfunctions, concludes this main part.

The last part of the paper deals with fractionalization without the necessity of diagonalization, by expressing the fractionalized operator in the form of a Fourier series. The procedure is applied again to the Fourier transformation, leading in a different way to the same results that were derived before.

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