Eigenfunctions of linear canonical transform

The linear canonical transform (the LCT) is a useful tool for optical system analysis and signal processing. It is parameterized by a 2/spl times/2 matrix {a, b, c, d}. Many operations, such as the Fourier transform (FT), fractional Fourier transform (FRFT), Fresnel transform, and scaling operations are all the special cases of the LCT. We discuss the eigenfunctions of the LCT. The eigenfunctions of the FT, FRFT, Fresnel transform, and scaling operations have been known, and we derive the eigenfunctions of the LCT based on the eigenfunctions of these operations. We find, for different cases, that the eigenfunctions of the LCT also have different forms. When |a+d|<2, the eigenfunctions are the scaling, chirp multiplication of Hermite functions, but when |a+d|=2, the eigenfunctions become the chirp multiplication, chirp convolution of almost-periodic functions, or impulse trains. In addition, when |a+d|>2, the eigenfunctions become the chirp multiplication and chirp convolution of self-similar functions (fractals). Besides, since many optical systems can be represented by the LCT, we can thus use the eigenfunctions of the LCT derived in this paper to discuss the self-imaging phenomena in optics. We show that there are usually varieties of input functions that can cause the self-imaging phenomena for an optical system.