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American Journal of Physics

Interference, diffraction, and refraction, via Dirac’s notation

F.J. Duarte

Published Jul 1, 1997• 24 citations
Publisher

Abstract

The diffraction grating equation and the law of refraction are derived, in the macroscopic domain, using a generalized interference equation derived via the application of Dirac’s notation to classical optics.
DOI
10.1119/1.18613

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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.
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.