THE LIMITATION OF THE NUMBER OF MAXIMA IN THE INTERFERENCE EXPERIMENT

Abstract

The work is devoted to the well-known problem in the wave theory of describing an interference experiment, which consists in the need to determine the spatial distribution of the maxima and minima of the wave field, which is a superposition of two coherent spherical waves. The paper discusses one essential moment of the theory of interference, which is not touched upon in the traditional presentation. The question concerns the number of maxima in the interference experiment.

It is shown that within the framework of the traditional approach, which is essentially approximate, namely the Fresnel approximation, the question of the number of maxima drops out by itself. This is due, in particular, to the fact that in the Fresnel picture, interfering spherical waves are replaced by paraboloid ones. Paraboloid waves, in turn, are replaced by plane waves in the Fraunhofer approximation. An exact expression for the difference between the paths of spherical waves is studied. An exact expression is obtained for the dependence of the positions of the maxima on the magnitude of the difference between the wave paths. It is shown that if the difference between the wave paths exceeds the distance between the sources, this expression takes on a complex value. It follows from this result that the difference between the wave paths cannot exceed the distance between the sources. The result of the limited value of the path difference is also reproduced in a vector way.

Based on the result on the boundedness of the path difference, it is shown that the number of maxima of the interference pattern is also bounded. The number of maxima is always less than the ratio of the distance between the source and the wavelength. If the wavelength exceeds the distance between the sources, one maximum should be observed in the interference experiment.