Topic 14. Wavefronts and Huygens’ Principle

 

          A wavefront is a surface over which an optical wave has a constant phase. For example, a wavefront could be the surface over which the wave has a maximum (the crest of a water wave, for example) or a minimum (the trough of the same wave) value. The shape of a wavefront is usually determined by the geometry of the source. A point source has wavefronts that are spheres whose centers are at the point source. A fluorescent tube would have wavefronts that are cylinders concentric with the tube itself. A very large sheet of material that is uniformly illuminated would generate wavefronts that are plane waves parallel to the sheet.

 

          The direction of propagation of the wave is always perpendicular to the surface of the wavefront at each point. Thus, the wavefronts of a point source are spheres and the wave propagates radially outward – the radius of a sphere is perpendicular to its circumference at each point. The same thing is true of the radius of the cylindrical wavefronts that would be generated by a fluorescent tube.

 

          Although the wave fronts produced by a point source are always concentric spheres in principle, when the source is very far away the radii of the spheres are so large that they look like plane waves to an observer. (Just as the Earth looks flat when viewed from a point near its surface.)

 

          Huygens’ principle (Christaan Huygens, 1629-1695, published about 1690) describes how a wavefront moves in space. According to this principle, we imagine that each point on the wavefront acts as a point source that emits spherical wavelets. These wavelets travel with the velocity of light in the medium. At any later time, the total wavefront is the envelope that encloses all of these wavelets. That is, the tangent line that joins the front surface of each one of them. A simple example of how a plane wavefront moves is shown below:

 

 

 

 

Wavefront after

 

Spherical wavelets

 

Wavefront before

 

 

The same construction is used for a wavefront of any other shape. When a wave travels in a single medium at a constant speed, the Huygen’s construction preserves the general form of the wavefront. That is, spheres propagate and become larger spheres, cylinders become larger cylinders, etc.

 

If a portion of the wavefront enters a different medium (enters glass from air, for example), then the wavelets generated by each portion of the wavefront travel with the velocity that is appropriate for the medium that the wavefront is in. That is, the wavelets in the medium where the speed of light is less will have smaller radii than the wavelets in the original medium.

 

          Although Huygens’ principle was initially stated without any proof, a slightly modified form of it was later (about 1815) derived by Fresnel from the mathematical theory of waves. Note that Huygens was a contemporary of Newton, and that it would probably have been much more difficult to publish his theory if he had lived in England, where disagreeing with Newton was not an easy or popular position to take.

 

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