Fresnel diffraction from polygonal apertures
Huang, JG, Christian, JM and McDonald, GS 2005, Fresnel diffraction from polygonal apertures , in: 25th European Quantum Electronics Conference (CLEO Europe/EQEC), 12th  17th June, 2005, Munich, Germany.

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Abstract
We present, for the first time, two complementary analytical techniques for calculating the Fresnel diffraction patterns from a polygonal aperture illuminated by a plane wave. These frameworks are exact, in that they do not involve any further approximation beyond the (paraxial) Fresnel integral. Here, we consider regular polygonal apertures, but our results are readily extended to describe nearfield diffraction from closed apertures of arbitrary shape. Our results are of fundamental importance and have specific applications where standard methods, such as Fast Fourier Transform (FFT) techniques, fail. For example, in unstableresonator mode calculations, both (paraxial beam) ABCD matrix modelling and existing semianalytical methods can only give accurate results in limited parameter regimes. Consequently, a complete and detailed study of optical fractal laser modes [1] has not previously been possible. A specific advantage of our formalisms is the ability to calculate and store the fine details of only a small portion of one, or many, complex diffraction patterns. Moreover, the explicit mathematical form of our results may also lend physical insight into a wide range of diffractionrelated phenomena in physics. For example, insight into the physical origin of excess quantum noise in lasers, where the transverse symmetry of an aperturing element has been shown to play a central role in the observed phenomena [2]. While Fraunhoffer (farfield) diffraction patterns have been known for many years, there has been relatively little equivalent published work in the Fresnel (nearfield)regime. The farfield approximation allows the expression of diffraction patterns and descriptions of derivative concepts (eg in holography, filtering, convolution and coherence) as simple Fourier integrals and transform theorems, respectively. Our new results permit the mathematical and physical expression of nearfield diffraction patterns in terms of their elemental spatial structures (edgewaves). It is plausible that our results may also open future doors in the development of derivative concepts in Fresnel Optics. The Fresnel diffraction patterns consist of a planewave (uniform) component, plus a rapidlyvarying interference contribution from boundarydiffracted waves from all edges of the aperture. This physical interpretation is present in the mathematical formulations of both the SFunction Method, which deals explicitly with edgewave combinations, and also the LineIntegral Method, where the Fresnel integral over the aperture area is expressed as a circulation around its edge. Extensive computational investigations have verified that our two approaches are completely equivalent, and produce identical results. The amount of fine detail present in the pattern can be quantified by the Fresnel Number N = a2/λL + f(n), where a is the radius of a circle inscribing an nsided polygon, L is the distance from the aperture to the observation plane (L^2 » a^2 is a paraxial approximation inherent to the nearfield approximation), λ is the wavelength of the illuminating light, and f(n) = 0.30618n2  0.19533n  0.68095 is a term allowing for the geometrical structure of the aperture. As n becomes larger, the level of detail increases. References [1] G.P. Karman, G.S. McDonald, G.H.C. New, and J.P. Woerdman, Nature 402 (1999) 138. [2] G.P. Karman, G.S. McDonald, J.P. Woerdman, and G.H.C. New, Appl Opt 38 (1999) 6874
Item Type:  Conference or Workshop Item (Poster) 

Themes:  Energy Media, Digital Technology and the Creative Economy Subjects outside of the University Themes 
Schools:  Schools > School of Computing, Science and Engineering > Salford Innovation Research Centre (SIRC) 
Refereed:  Yes 
Depositing User:  JM Christian 
Date Deposited:  17 Oct 2011 11:31 
Last Modified:  29 Oct 2015 00:12 
URI:  http://usir.salford.ac.uk/id/eprint/18429 
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