Monday 8-5-96
The relevant section in the book is 23.1
Homework problems : 3, 9, 18
Chapters 21 and 22 dealt with reflection and refraction, which can be analyzed using geometrical optics, a simple model of light that uses rays and wave fronts. In chapter 23 we'll get away from that simple model, and analyze situations based on physical optics, which treats light as a wave with a speed, wavelength, and frequency. We need to do this because geometrical optics breaks down (i.e., doesn't give us anything close to the right answers) when dealing with situations when light interacts with things (thin slits, thin films, etc.) which are small in size, of the order of the wavelength of light.
Physical optics, then, accounts for the wave nature of light. If we're going to examine the wave nature of light, a quick review of waves is a good idea. One wave property is interference, which may be constructive or destructive. If you take two waves and bring them together, they will add wherever a peak from one matches a peak from the other. That's constructive interference. Wherever a peak from one wave matches a trough in another wave, however, they will cancel each other out (or partially cancel, if the amplitudes are different); that's destructive interference.
Light, because it exhibits wave properties, will show this wave interference. This was first shown in 1801 by Thomas Young, who sent sunlight through two narrow slits and showed that an interference pattern could be seen on a screen placed behind the two slits. The interference pattern was a set of alternating bright and dark lines, corresponding to where the light from one slit was alternately constructively and destructively interfering with the light from the second slit.
You might think it would be easier to simply set up two light sources and look at their interference pattern, but the phase relationship beteen the waves is critically important when it comes to interference, and two sources tend to have a randomly varying phase relationship. With a single source shining on two slits, you ensure a constant phase difference (or even, preferably, no phase difference at all) between the light from the two slits.
This makes use of Huygen's principle, the idea that each point on a wave can be considered to be a source of secondary waves. Applying this to the two slits, each slit acts as a source of light of the same wavelength, with the light from the two slits interfering constructively or destructively to produce an interference pattern of bright and dark lines.
For two slits separated by a distance d, and emitting light at a wavelength lambda, light will constructively interfere at certain angles. These angles are found by applying the equation:
dsin theta = n lambda
where n is an integer. Putting this in words, you get constructive interference when the path difference to a point from the two slits is an interger multiple of the wavelength. Which makes sense...if two waves of the same wavelength are shifted by 1, 2, 3, etc. wavelengths, the peaks and troughs will still all line up, giving constructive interference. n actually represents the number of wavelengths one wave is shifted relative to the other.
If the interference pattern was being viewed on a screen a distance L from the slits, the wavelength can be found from the equation:
lambda = y d/nL
y here is the distance from the center of the interference pattern to the nth bright line in the pattern. That applies as long as the angle is small, at least (y small compared to L).