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Eigenfunctions of the canonical transform and the self-imaging problems in optical system

The affine Fourier transform (AFT) also called as the canonical transform. It generalizes the fractional Fourier transform (FRFT), Fresnel transform, scaling operation, etc., and is a very useful tool for signal processing. We derive the eigenfunctions of the AFT. The eigenfunctions seems hard to be derived, but since the AFT can be represented by the time-frequency matrix (TF matrix), we can use the matrix operations to derive its eigenfunctions. Then, because many optical systems can be represented as a special case of the AFT, the eigenfunctions of the AFT are just the light distributions that will cause the self-imaging phenomena for some optical systems. We use the eigenfunctions we derive to discuss the self-imaging phenomena.