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Huygens' Principle

At the beginning of this chapter we pictured only PLANE WAVES, in which the wavefronts ("crests" of the waves) form long straight lines (or, in space, flat planes) moving along together in parallel (separated by one wavelength $\lambda$) in a common direction $\Hat{k}$. One good reason for sticking to this description for as long as possible (and returning to it every chance we get) is that it is so simple - we can write down an explicit formula for the amplitude of a plane wave as a function of time and space whose qualitative features are readily apparent (with a little effort). Another good reason has to do with the fact that all waves look pretty much like plane waves when they are far from their origin.^14.21We will come back to this shortly. A final reason for our love of plane waves is that they are so easily related to the idea of "RAYS."

In GEOMETRICAL OPTICS it is convenient to picture the wavevector $\Vec{k}$ as a "ray" of light (though we can adopt the same notion for any kind of wave) that propagates along a straight line like a billiard ball. In fact, the analogy between $\Vec{k}$ and the momentum $\Vec{p}$ of a particle is more than just a metaphor, as we shall see later. However, for now it will suffice to borrow this imagery from Newton and company, who used it very effectively in describing the corpuscular theory of light.^14.22

However, near any localized source of waves the outgoing wavefronts are nothing like plane waves; if the dimensions of the source are small compared to the wavelength then the outgoing waves look pretty much like SPHERICAL WAVES. For sources similar in size to  $\lambda$,  things can get very complicated.

Christian Huygens (1629-1695) invented the following gimmick for constructing actual wavefronts from spherical waves:

HUYGENS' PRINCIPLE:
 

This may be seen to make some sense (try it yourself) but its profound importance to our qualitative understanding of the behaviour of light was really brought home by Fresnel (1788-1827), who used it to explain the phenomenon of diffraction, which we will discuss shortly. But first, let's familiarize ourselves with the simpler phenomena of interference.

Footnotes

. . . origin.^14.21

This is sort of like the mathematical assertion that all lines look straight if we look at them through a powerful enough microscope.

. . . light.^14.22

"Corpuscles" are hypothetical particles of light that follow trajectories Newton called "rays," thus starting a long tradition of naming every new form or radiation a "ray."