Optical wave propagation of fractal fields

The evolution of the complex amplitude of a fractal optical field during propagation through free space (Fresnel transform) and through a quadratic refractive index medium (fractional Fourier transform) is discussed. To that end, the two above transforms have been introduced within the framework of the generalized Fresnel transform (GFT) and their scaling properties formulated. These scaling properties are shown to be suitable to reveal the self-similarity behaviour of the fractal field. A general method to determine the fractal dimension and other fractal features from the diffraction patterns is proposed. As an application of the method, a numerical experiment has been carried out for the diffraction of a triadic Canter set in free space and in a quadratic index medium. The determination of the fractal dimension is illustrated in the two cases.