Multiple slit interference: a hyperbola based analysis

Abstract

Abstract The textbook description of multiple slit interference employs a standard path difference formula that was originally calculated for the purpose of analyzing Young’s double slit experiment. It was arrived at on the basis of a pair of assumptions that help simplify the geometry of the slit-barrier-screen arrangement and the ensuing formalism needed to estimate the positions of interference fringes on the detection screen. According to the conventional approach, convergent rays of light that emanate from different slits can be treated as effectively parallel in the far field limit. Such a parallel ray approximation-based analysis was shown by Thomas (2019; 2020) to be redundant, paradoxical and limited in accuracy. In this paper, the task of reformulation that began previously with the elementary two-slit experiment is now extended to encompass the N-slit scenario, diffraction at a single-slit and the formation of circular fringes. In order to derive one of the key results (the generalized hyperbola theorem for N-point sources), a novel computational technique called analytical induction is introduced. The procedure involves a synthesis of concepts from Cartesian geometry and discrete mathematics, namely coordinate transformation by Euclidean translation and a method of proof called the principle of mathematical induction, respectively. The final goal attained are two distribution functions that succinctly and precisely describe the variation of intensity of interference fringes on the distant screen, when it is oriented parallel and perpendicular relative to the line joining the sources. A comparison is drawn between the predictions of the conventional and the new analyses by means of numerical-graphical simulation with MATLAB. The theoretical methods and results presented herein may have a significant bearing in areas of applied optics like interferometry and diffraction crystallography. On the pedagogic front, it is suggested that the current geometrical program being visually more intuitive, be incorporated into the standard curriculum of an advanced undergraduate/graduate level course in physical optics, to complement the conventional approach.