# Diffraction of fractal light: New frontiers for the mathematics of edge waves

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## Abstract

The diffraction pattern produced by a plane wave (i.e., a perfectly uniform wavefront) scattering from an infinite hard edge is well-described by the Fresnel integral [1]. Such one-dimensional (1D) edge waves [see Fig. 1(a)] turn out to be truly elemental spatial structures in linear optical systems in the sense that patterns produced by other apertures [such as a slit – see Fig. 1(b)] can be decomposed into a sum of two interfering edge waves. Our group has previously established that such waves also play a fundamental role in the exact mathematical description of diffraction patterns generated from uniform illumination of polygonal apertures [2], whereby one superposes the waves from all constituent edges (each of which has, crucially, a finite length). Here, we report on the first steps taken toward considering a related but distinct physical problem, namely how a fractal light wave incident on an infinite edge is diffracted in both the near and far fields. Our method is based upon a Fresnel-type prescription, generalizing earlier analyses [1,2] to accommodate an illuminating field that comprises a spectrum with many distinct components (each spatial frequency contributes a characteristic scale length to the incident pattern). Our results can be readily applied to other classic 1D and 2D systems such as slits and polygons, respectively. References [1] M. P. Silverman and W. Strange, Am. J. Phys. 64, 773 (1996). [2] J. G. Huang, J. M. Christian, and G. S. McDonald, J. Opt. Soc. Am. A 23, 2768 (2006).

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