Serrated Circular Apertures: Optical Fourier Transforms and Fractal Analysis.

The optical transform of the transmission function of a serrated circular aperture is discussed, with particular emphasis on the relationship between the features in the transform and the parameters that describe the aperture serration. The transform is produced in a canonical optical processing system, where the scalar field distribution in the back focal plane of the lens is proportional to the two-dimensional spatial Fourier transform of the aperture transmission function. In the statistical diffraction theory, the quantity of interest is the two-point moment of the intensity, which is a fourth-order moment of this scalar field component. A careful calculation of the diffracted field is performed. The two-point intensity moment is expanded in terms of second-order moments of the field. Due to the polar symmetry of the field, circularity does not hold and the two significant terms in the expansion are identical but for a pi rotation. From the detailed expression for the remaining second-order moment, interesting features of the optical transform are extracted. These features are ring fragmentation, the number of transform spikes, and spike appearance, which correspond to serration roughness, correlation angle, and correlation function, respectively. The results of computer simulations and optical experiments support the predicted relationships between the aperture parameters and transform features. Detailed study and modeling of the errors introduced during fabrication of apertures for use in the experiments show that very small errors (~2 mu m) are easily seen in the optical Fourier transform. The effects of the variations in the parameters of the serration on the fractal dimension of the aperture is also discussed. Geometric techniques are used to measure the fractal dimension of computer-designed apertures. The fractal dimension depends on the roughness sigma and correlation length L of a serrated aperture or edge by way of the ratio sigma/L for a given correlation function shape. Changing the correlation function alters this dependence: the fractal dimension increases when the function is sharpened.