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European Journal of Physics

Comment on Reflection, refraction and multislit interference

F.J. Duarte

Published Jul 9, 2004• 13 citations
Publisher

Abstract

It is indicated that the generalized interference equation, obtained using the Dirac notation applied to an N-slit interferometer, has already been applied to derive the equation of the diffraction grating, the law of refraction and the law of reflection. The applicability of the Dirac approach to describe interference of large number of indistinguishable photons is also explained.

DOI
10.1088/0143-0807/25/5/l04

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It is indicated that the generalized interference equation, obtained using the Dirac notation applied to an N-slit interferometer, has already been applied to derive the equation of the diffraction grating, the law of refraction and the law of reflection. The applicability of the Dirac approach to describe interference of large number of indistinguishable photons is also explained.
The interaction of coherent light with a set of slits, in the near field, gives rise to an interference pattern. Refraction is the change in the geometrical path, of a beam of light, due to transmission from the original medium of propagation to a second medium with a different refractive index. One of the beauties of the Dirac description of optics is the ability to continuously describe the evolution of the interferometric distribution, as it moves from the near to the far field, with a single mathematical equation. The interferometric distribution, in the near field, is mainly part of a single order. At the boundaries there is an incipient indication that of emerging orders. Explicitly, if a mirror is placed at an infinitesimal distance immediately behind the N-slit array then the transmission interferometer becomes a reflection interferometer.
Quantum interference, which is a physical phenomenon, and probability amplitudes à la Dirac are miraculously like synonyms. Indeed, the way probability amplitudes are written down on paper is very much as a single photon, or an ensemble of indistinguishable photons, propagates from one plane to the next. And then, if at one of the planes an N -slit array alternative is offered then the probability amplitude notation à la Dirac cleverly describes the N -path alternative via a mathematical summation. Here we should emphasize that since the photon is a nonlocal quantum entity, all the microscopic slits in the N -slit array, or transmission diffraction grating, are illuminated simultaneously.
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.
It is indicated that N-slit quantum interference can be correctly and completely described via the wave functions of ‘ordinary wave optics’ à la Dirac. Here, an interferometric outline is given that includes the measurement process at the detector. This approach utilizes a description of the photon as a nonlocal wavy form of energy. It is emphasized that the spatial features of N-slit interferograms are nicely explained by standard Diracian quantum mechanics without the need to introduce new theoretical states of light.