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American Journal of Physics

An exactly soluble Fresnel diffraction model of two-slit interference

James A. Lock

Published Oct 1, 1996• 1 citation
Publisher

Abstract

Using the formalism of Fresnel diffraction, we examine the diffraction and interference of two initially widely separated parallel Gaussian light beams. This two-beam configuration is a version of Young’s two-slit interference problem that is free of the mathematical complexity that occurs in most Fresnel diffraction calculations. As a result, it clearly and simply illustrates the transition from ray optics to wave interference.

DOI
10.1119/1.18375

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•Mar 15, 2024•4 citations
Using the formalism of Fresnel diffraction, we examine the diffraction and interference of two initially widely separated parallel Gaussian light beams. This two-beam configuration is a version of Young’s two-slit interference problem that is free of the mathematical complexity that occurs in most Fresnel diffraction calculations. As a result, it clearly and simply illustrates the transition from ray optics to wave interference.
Using the Fresnel–Kirchhoff integral, we express the state of the light that diffracts from an aperture to a point, by superposition of two or more waves. The amplitudes of the waves are determined by the distances of the point from the ray optics borders of the emerging beams. The interference approach to the Fresnel diffraction leads to a universal formulation of the normalized intensity distribution on the diffraction pattern that determines the state of the incident light, the parameters of the aperture, and its distance from the observation plane, uniquely. In the photon approach, according to the uncertainty principle, confining the position of a photon to an aperture changes the probable propagation direction of the photon in an interval that leads to the diffraction of the photon or light. Thus, the normalized intensity distribution on the diffraction pattern is the probability distribution of a photon. To this probability distribution, a probability amplitude distribution is associated that diffracts similar to a wave in Fresnel diffraction. Applying the introduced approach to the study of the diffraction, from different simple apertures and phase steps, we deduce general behaviors of the Fresnel diffraction and show it is involved in all aspects of light phenomena.
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.
Young’s double slit experiment was called as "a phenomenon […] has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot make the mystery go away by ‘explaining’ how it works". To study the mystery, we perform the non-crossing-slit-experiments and crossing-slit experiments and show phenomena: (1) in the same double slit or cross double slit experiments, the non-interference patterns gradually evolve to the interference patterns; (2) in the same single slit or cross single slit experiments, the non-diffraction patterns gradually evolve to the diffraction patterns; (3) in the same two-non-parallel-slits experiment, the non-wave patterns gradually evolve to the hybrid pattern (i.e., interference-pattern-embedded-in-diffraction-pattern); (4) in the same curved-double slit experiments, the non-interference patterns evolve to Seashell-shape interference patterns. The above phenomena show new mysteries that are more mysterious than that of the double slit experiments: it is challenge to consistently explain how they work. Those novel phenomena indicate that the standard theories of the double slit are incomplete. Novel experiments bring the search for the new theory of physical/quantum optics.
We show, both analytically and experimentally, that under common experimental conditions the interference pattern produced in a classic Young's double-slit experiment is indistinguishable from that generated by means of a doubly refracting uniaxial crystal whose optic axis makes a skew angle with the light propagation direction. The equivalence between diffraction and crystal optics interference experiments, taken for granted by Arago and Fresnel in their pioneering research on the interference of polarized light beams, is thus rigorously proven.
Young’s double slit experiment was called as "a phenomenon […] has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot make the mystery go away by ‘explaining’ how it works". To study the mystery, we perform the non-crossing-slit-experiments and crossing-slit experiments and show phenomena: (1) in the same double slit or cross double slit experiments, the non-interference patterns gradually evolve to the interference patterns; (2) in the same single slit or cross single slit experiments, the non-diffraction patterns gradually evolve to the diffraction patterns; (3) in the same two-non-parallel-slits experiment, the non-wave patterns gradually evolve to the hybrid pattern (i.e., interference-pattern-embedded-in-diffraction-pattern); (4) in the same curved-double slit experiments, the non-interference patterns evolve to Seashell-shape interference patterns. The above phenomena show new mysteries that are more mysterious than that of the double slit experiments: it is challenge to consistently explain how they work. Those novel phenomena indicate that the standard theories of the double slit are incomplete. Novel experiments bring the search for the new theory of physical/quantum optics.