Collins formula in frequency-domain described by fractional Fourier transforms or fractional Hankel transforms

The Collins formula in frequency-domain for description of angle spectrum transformation through ABCD optical systems can he written in terms of a fractional Fourier transform in the Cartesian-coordinates, or can be written in terms of a fractional Hankel transform in the cylindrical coordinates. It is shown that, when the matrix element D = 0, it is related to the standard Fourier transform or standard Hankel transform for angle spectrum; and when C = 0, it corresponds to image-forming operator transform in frequency-domain. The case of a thin lens is discussed as also a special one.