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Armenian Journal of Physics

Description of the Classical Diffraction Experiment of a Systemof Many Slits and the Interference Pattern of Two Sources within the Framework of a Single Scheme

А. Ж. Хачатрян, Zh. R. Panosyan, А.Ф. Парсамян, L.R. Soghomonyan

Published Sep 4, 2023• 0 citations
Publisher

Abstract

The problem of an approximate description of the wave field of many sources is considered. It is shown that, if the problem is considered more comprehensively, the description of the interference and diffraction experiments can be given within the framework of a single scheme. It is shown that the character of wave amplification in these two classical experiments is different. In the case of an interference experiment, the field amplification is fixed at a point, while in the case of a diffraction experiment the field amplification is fixed in the observation direction. The problem of determining the intensity maximum of a diffraction grating in the Fresnel pattern is studied under the condition that the maximum is satisfied in the Fraunhofer pattern. A condition under which a diffraction grating behaves like a lens is obtained.

DOI
10.54503/18291171-2023.16.2-40

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The problem of an approximate description of the wave field of many sources is considered. It is shown that, if the problem is considered more comprehensively, the description of the interference and diffraction experiments can be given within the framework of a single scheme. It is shown that the character of wave amplification in these two classical experiments is different. In the case of an interference experiment, the field amplification is fixed at a point, while in the case of a diffraction experiment the field amplification is fixed in the observation direction. The problem of determining the intensity maximum of a diffraction grating in the Fresnel pattern is studied under the condition that the maximum is satisfied in the Fraunhofer pattern. A condition under which a diffraction grating behaves like a lens is obtained.
Abstract In Chapter 21, we turn to interference and diffraction. We start by describing the interference pattern in Young’s classic two-slit experiment, whose analysis is subsequently generalised to determine the interference pattern behind a multi-slit grating. The diffraction pattern for a single slit is then estimated using Huygens’ principle, and this is combined with the interference pattern for a multi-slit grating to get the entire wave pattern generated by a diffraction grating. Next, we turn to the diffraction of electrons. This starts with a brief introduction to quantum mechanics, focussed on Schrodinger’s wave equation, de Broglie waves, and the collapse of the wavefunction following measurements. Finally, we use these elementary quantum concepts to describe the diffraction of electrons.
Interference and diffraction of light are the nature performance of wave character,which is the result of superposition of waves.The essential difference between interference and diffraction phenomena is that interference comes from the superposition by the two or limited interfering beams of light,and diffraction is from the superposition by an infinite beam.From the perspective of optical path difference,corresponding to the same optical path difference,the minimum conditions of single-slit diffraction becomes the maxima conditions of double-slit interference.The reason is that the slot width is so small that can be considered as a single light source under the double-slit case,and the interference in the same slit can be ignored;while the single-slit diffraction minimum conditions are exactly correspond to the interference in the same slit.
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.
This paper provides a complete physical model associated with mathematical analysis and formulation that is based on Schrödinger equation and Feynman path integral. This model answers to single Photon (electron) double slit experiment with high degree of accuracy. It also gives a new method of analysis and formulation for Fresnel and Fraunhofer diffractions that are completely according to the Schrödinger equation and Feynman path concept. The results of these methods are precisely in agreement with the near field and far field diffraction experiments. In other words the new formulation in all above mentioned phenomena i.e. single photon (electron) double slit experiment, Fresnel and Fraunhofer diffractions are completely confirmed by lab experiments(measurements). Lab experiments mean common intensity and pattern experiments resulting from Fresnel (near field) diffraction, Fraunhofer (far field) diffraction and patterns resulting in single photon (electron) lab experiment. The resultant formulas are sketched and calculated by matlab program, the resultant graphs and values are compared with lab experiments in each section. This analysis substitutes wave function and probability density function concepts instead of interference of light beams emitted from infinitesimal pinholes or classical interpretation of near field diffraction and far field diffraction phenomena. The ultimate goal of this paper is to give a very accurate answer to the single particle double slit experiment and also to unify diffraction concept of classical optics in to quantum mechanics, deleting the traditional concept of phases in classical optics and substituting the concept of deflection of Feynman paths. The above mentioned analysis may be a glad tiding to a reliable method to unify classical optics in to quantum mechanics.
In analogy to the well-known diffraction of waves at spatial structures, one can also define diffraction in time when a wave is temporarily modulated. In the present paper we investigate this phenomenon starting from a Green's-function approach. Diffraction in time appears for Schr\"odinger waves and not for light in vacuum. Specific cases of diffraction in time investigated here are diffraction at an ``edge in time'' and a Fresnel lens in time. We then investigate in detail the simultaneous diffraction both in space and in time representing a general solution. Then we analyze in detail diffraction at an edge, both in space and in time, of a single slit in space and in time, a double slit in space, and a single slit in time, a single slit in space and a double slit in time, and, finally, a double slit both in space and in time. In all cases we analyze the possibilities of various approximations which can be made, and show the limits and validity of the Fraunhofer approximation both in space and in time. We give explicit results for a gedanken experiment with very cold atoms.
Current nanotechnology techniques make possible the preparation of slits in the submicrometer range so that electron interference and diffraction experiments can be done even with a conventional electron microscope. If the instrument is also equipped with a field emission source, it is possible to follow almost in real time the transition from the image of the slits to their Fraunhofer pattern through the intermediate Fresnel diffraction images. We discuss our results for the two-slit experiment and illustrate them for the three-slit case.