# FisMat2017 - Submission - View

**Abstract's title**: Sharp-edge diffraction theory revisited: a catastrophic perspective

**Affiliation**: Università degli Studi "Roma Tre"

**Affiliation Address**: Dipartimento di Ingegneria Via Vito Volterra, 62 I-00146 Rome

**Country**: Italy

Fresnel’s diffraction theory is a milestone of classical optics since two centuries.

Lesser-known is that several decades before Fresnel’s ideas rose to such a paradigmatic role, Thomas Young had already proposed his own interpretation of diffraction phenomena. Young’s idea was that diffraction is the result of the superposition of geometrical rays leaving the aperture and wavelets generated at the aperture rim.

Young’s picture was not adequately supported from a mathematical viewpoint, and the more predictive Fresnel theory of light prevailed. However, between the end of the 19th and the beginning of the 20th centuries, Gian Antonio Maggi and, independently, Wojciech Rubinowitz gave Young’s ideas the status of a quantitative theory, the so-called boundary diffraction wave (BDW) theory.

More than one century later, people continue to consider BDW's theory only a mere computational alternative to the ``orthodox'' diffraction theory. What we have found is that BDW’s theory, if reformulated from a suitable point of view, reveals an unexpectedly powerful tool for understanding sharp-edge diffraction. Such ``unorthodox’’ perspective is provided by the so-called *Catastrophe optics* (CO), introduced at the end of seventies by Michael Berry and John Nye as a new and modern theoretical framework aimed at dealing with the so-called *natural focusing* of light. Rather than solving wave equations, CO’s mathematical description of wavefields is built onto skeletons of bright caustics, representing the geometrical optics limit. To quote Nye's own words, the aim of CO is just to ``to add diffraction to the caustics of ray optics’'. Nye's thought fits perfectly Young’s diffraction philosophy.

The present work is based on a paraxial development of the BDW theory proposed in 2000 by John Hannay. Our analysis consists ultimately in a nontrivial saddle-point treatment of Hannay's phase integral, carried out within CO’s prescriptions [1]. In particular, a key role is played by a geometrical interpretation of the BDW saddles. Through it, the latter are suitably ``imaged’’ onto the diffracting aperture rim. In this way it is possible to relate the focusing of the boundary wavelets, that generates the caustics topology, with the ``birth’’ or the ``death’’ of one or more saddles. This new perspective on diffraction gave renewed potential interests to some old experiments, today nearly forgotten [2]. More importantly, it allows unexpected diffraction phenomena to be described and interpreted in a way that could hardly be obtained in terms of Fresnel’s theory. The generation of pseudo-nondiffracting wavefields as well as the exploration of 2D and 3D optical singularities are only two examples.

[1] R. Borghi, *Catastrophe optics of sharp-edge diffraction*, Opt. Lett. **41**, 3114 (2016).

[2] R. Borghi, *Heart diffraction*, Opt. Lett. **42**, 2070 (2017).