Diffraction by a Fractal Transmittance Obtained from a Multiplicative Superposition of Periodical Rectangular Functions

Abstract

The Fraunhofer region of one-dimensional fractal diffraction grating is studied when this grating is defined using a set of scaled periodical (and rectangular) functions. The fractal structure is obtained through a Cantor density function, which is defined from an incremental product of such periodical components. Each of these components can be filtered individually and so, the corresponding intensity pattern can be evaluated. A mathematical foundation for such decomposition is developed using the concept of characteristic (or indicator) function of a set. Graphic results are shown and the partial self-similarity of lateral intensity distribution calculated in any transversal plane to the optical axis for a certain value of magnification. With these results, fractal, periodical and intrinsic periodical domains can be distinguished to characterize the intensity patterns. The self-similarity of intensity distributions and the contrast of that function are calculated for different states of the Cantor transmittance.