16 - The stationary phase approximation

Summary

Introduction

In this chapter we begin to forge a connection between the ray theory and the wave theory of light, two topics that so far have been treated as entirely separate and disconnected. Fresnel showed in the early nineteenth century how the idea of straight line propagation can be reconciled with the wave theory by using what are now called Fresnel zones. His reasoning went as follows. A source point S radiates a spherical wavefront towards a circular aperture, as shown in fig. 16.1. To find the amplitude of the light wave at a point P beyond the aperture, each point in the part of the wavefront not stopped by the screen may be considered as a secondary source. The amplitude at P is the sum of the amplitudes contributed by each of these secondary sources. In this summation the relative phase of the contributions plays a crucial role.

To get a handle on the summation, Fresnel divided the wavefront into annular zones. These zones are bounded by circles, chosen such that successive distances SQ₁P, SQ₂P, SQ₃P… differ by half a wavelength. It is not difficult to show that the areas of the zones so constructed are very nearly equal. So the waves arriving at P coming from two adjacent zones have the same amplitude, but a phase difference of 180° on account of the way in which the zones were constructed. The contributions from adjacent zones therefore cancel each other.