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Huygens' principle and the wave model

Page history last edited by Matt Harrington 9 years, 12 months ago Saved with comment

Working Content  > The wave model

 

Prerequisites:

 

Back in the 17th century, when Newton was making great strides in understanding the nature of light with his model of light as small, very fast moving particles, a Dutch competitor, Christian Huygens, had another idea: light was an oscillation, like sound or water waves. Unfortunately for Huygens, his model was more complicated than Newton's to calculate with, it didn't do any better than Newton's model (for anything that could be measured -- at the time), and besides, Newton had the star billing. Not much attention was paid to his model (though it had its advocates) until a fateful competition more than 100 years later.

 

Young's experiment and the French Academy Prize

At the end of the 18th century (1799), an English scientist, Thomas Young, began reviving Huygens' wave model. His two slit experiment, published in 1803 (discussed briefly below and discussed in more detail in the page, Two slit interference) was very hard for the particle model to explain. More people became interested in the wave model and, in 1817, the French Academy of Sciences, proposed a competition for papers on the theory of light.

 

Most of the academy members supported the particle model and hoped to kill the wave theory once and for all. But French physicist, August Fresnel, presented a paper with a detailed wave theory that permitted calculations to be done. The academy members pounced. Some noticed right away that his theory predicted a  strange result. If a point source of light were blocked by a perfect circular disk it would produce a circular shadow -- but Fresnel's version of the wave theory predicted that there should be a bright spot at the center of the shadow. (To see why -- after you have read through the pages on the wave theory and understanding interference, go to the page on Arago's bright spot.) Of course, you have already figured out what happened. Despite the scoffers in the academy, one of them decided the experiment had to be tried. To almost everyone's surprise, the spot was indeed found to be there. This was enough to convince most scientists that the wave theory had to be right and Fresnel was awarded the prize. The wave theory was assumed to be correct -- until a century later, when Einstein came along and things became a lot more complicated!

 

Young's two slit interference

From our study of the ray model and our understanding of the patterns of light and shadow, if we set up a situation light shown at the right, with a tiny (point source) bulb, a mask with a small slot cut out of it, and a screen on which to observe the transmitted light, we expect, in the ray model, to see a bright spot that has the shape of the slot in the mask. We can easily see this by taking a spray of rays from the bulb in all directions. Those that are not blocked by the mask will just go straight through, painting a copy of the slot in the mask in light on the screen, as shown.

 

But now what would happen if we made the slot on the screen narrower? Surprisingly, once the slot gets pretty narrow (less than a mm), we begin to notice that there is light outside the shadow. In fact, as the slot gets narrower, the spot on the screen begins to spread out! What's happening?

The particle modelers suggested that perhaps as the slot got narrower, that the particles of light were scattering off the edges of the slit. (WARNING: This is NOT what is happening here!) But Thomas Young came up with another, even more perplexing experiment: one with two slits.

Young's experiment is shown in the figure at the right.  When the slits are narrow, what happens? What the particle theory predicts is that the pattern observed for the two slits will be the sums of the patterns that were observed for the individual slits, shifted a bit. After all, in the particle model, all that can matter is the total number of particles received and scattering from a point on the screen.

 

But what happens is something very different. The actual pattern for one and two slits (taken with a laser source, not a yellow bulb) looks like the figures below.

 

 

The extraordinary fact about this result is that there are some places where adding more light by opening another slit makes the result darker, not brighter. This really is a fatal flaw in the particle model. If the intensity is the total number of light particles, there is no way that adding more light (from a second source) can make anything darker. This suggests:

 

Light must somehow be both positive and negative and that two sources of light can cancel each other -- at least in some localized spots.

 

This was a serious challenge to the particle model and the central spot result was the final nail in the coffin. The wave theory won. (At least for a century.)

 

Starting simple: The water wave analogy

We are going to see that "what is waving" in light are in fact electromagnetic fields. But these are rather difficult to get a picture of, especially since they propagate in three dimensions. We began our discussion of waves with a one-point-particle example: the mass on a spring. We then extended to a one-dimensional example (corresponding to infinitely many one-point particles): waves on an elastic string. Rather than leap right away into 3D, lets try to see if we can make sense of a two-dimensional example: waves on the surface of water. The mechanism is not quite the same as our 1D wave on an elastic string. The water wave is indeed transverse, mostly, but the water particles really move in little vertical circles, not just perpendicular to the surface of the water. We will ignore this complication here and treat it as if the water waves are moving up and down. (Our point is not to describe the motion of waves on water, but to construct a conceptual model that helps in thinking about the complex superpositions that happen with waves in 2 and 3D.)

 

If you drop a small stone in a pond, it will send out a circular ripple -- a propagating pulse in 2D. If we do it multiple times we will get propagating circular rings -- a string of pulses. If we put a rod into the water driven up and down by a rotating wheel, we will get sinusoidal circular waves moving outward.

 

Just as each bead in our model of the elastic string moves up and down to drive the next bead along in the same way as the initiating driver did the first bead, each bit of water moving up and down serves as a source for outgoing circular waves on the surface of the water. This is the key to Huygens' wave model.


PhET simulation. Click the image to run.

 

The foothold principles of Huygens' wave model

Whatever it is that's waving -- water, sound, or light --Huygens' key idea is the wavefront. For one pulse, it makes sense to think about it as the circle (in 2D -- or sphere in 3D) of equal amplitude that propagates outward from the source. For sinusoidal waves, it is more useful to think of the wavefront as lines (in 2D) or surfaces (in 3D) of equal phase. This means the points at which the angle in the argument of the sine function stays the same. So if we have a spherical sinusoidal wave that looks like sin(kr-ωt) = sin{k(r-vt)}, then the phase stays the same when r-vt stays constant; so as t increases, r grows at a speed v. The wavefront moves outward with the wave speed.

 

Here are the basic ideas of the model.

 

  • Each point on a wavefront acts like a source for a new spherical wavefront going out. These secondary waves are referred to as wavelets. (We need them here whereas we didn't on the elastic string since we are in 2 or 3D. As a result, there are many sources of waves moving on, not just one. This is what makes the Huygens' theory complex and hard to calculate. Calculating a result typically involves doing a complicated integral in 3D. This is what Fresnel figured out to win the French Academy Prize in 1817.)
  • The sum of the wavelets from the wavefronts at one time creates new wavefronts at future times.
  • The speed of the wavelets depends on the medium, not on the amplitude. (Though for some kinds of waves and some kinds of media, the speed of the waves can depend on the wavelength. Such systems are called dispersive.)

 

The Huygens' model for the case of light

In the case of light, there are some interesting results.

 

  • In the Huygens' model, rays are interpreted as the lines that are perpendicular to the wavefronts of the model.
  • If we assume that light propagates more slowly in dense media (in contrast to Newton's model), the Huygens' model can derive both the mirror rule and Snell's law can be derived using just geometry. (See the page Huygens and the basic results of the ray model (technical).)
  • Multiple waves from different sources pass through each other and just add their values to get a result. Superposition holds, just as it did for pulses on an elastic string (and as it does for electric fields).

 

These results mean that Huygens' model can not only reproduce the results of the ray model, but they also allow for interference -- having more sources can result in cancellations at particular points, not just enhancements.

 

In the follow-ons, work through the examples of how the Huygens' model correctly predicts the interference pattern of light at two slits, and even the diffraction pattern (spreading out) of light from a single narrow slit.

 

Follow-ons:

 

Joe Redish 4/23/12

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