Two source interference

Interference

Waves on a pond:

Think of when you drop a pebble into a pond, you will see circular waves eminate from the point where you dropped the pebble.

Two point sources

Lets take a particular example of two point sources separated by a distance d. The waves emitted by point source are spherical and thus can be written $$\vec{E}=\frac{{\vec{E}}_0}{r}\cos (kr-\omega t)$$ To make the problem easier we will make the $k$ 's the same for the two sources. Also lets set the $E_0$ 's to be the same as well.

Clearly I will be a maximum when the cosine is = +1 $$k\Delta r=2n\pi n=0,1,2\dots$$ $$\frac{2\pi }{\lambda }\Delta r=2n\pi$$ $$\Delta r=n\lambda$$ There will be a minimum when the cosine is = -1 $$k\Delta r=n\pi n=1,3,5\dots$$ $$\Delta r=\frac{n\lambda }{2}n=1,3,5\dots$$ So you get light and dark bands which are called interference fringes.To reiterate; we have two rays of light eminating from two point sources. Wehave looked at the combined wave at some point, a distance $r_1$ from the first source and a distance $$ from the second source. In that case we find that the intensity is proportional to $\frac{1}{R^2}{E}_0^2(1+\cos k\Delta r)\text{.}$ To make things easier we can redefine $E_0$ to be the amplitude of the waves at the point under consideration, that is $I={\epsilon }_{0}c{E}_0^2(1+\cos k\Delta r)\text{.}$ Or we can say $I_0=\epsilon c{E}_0^2/2$ and write $I=2I_0(1+\cos k\Delta r)\text{.}$

Say we place a screen a distance S away from the two sources:

Young's double slit

Young's double slit.is an excellent example of two source interference. The equations for this are what we worked out for two sources above. Interferenceis an excellent way to measure fine position changes. Small changes in $\Delta r$ make big observable changes in the interference fringes.

Michelson interferometer

A particularly useful example of using interference is the Michelson interferometer. This can be used to measure the speed of light in a medium,measure the fine position of something, and was used to show that the speed of light is a constant in all directions.

What really matters is the change in the optical pathlength. For example you could introduce a medium in one of the paths that has a different index ofrefraction, or different velocity of light. This will change the optical pathlength and change the interference at the observer. Thus you can measurethe velocity of the light in the introduced medium.

Michelson and Morely used this technique to try to determine if the speed of light is different in different directions. They put the whole apparatus on arotating table and then looked for changes in the interference fringes as it rotated. They saw no changes. In fact they went so far as to wait to see whathappened as the earth rotated and orbited and saw no changes. They thus concluded that the speed of light was the same in all directions (which nobodyat the time believed, even though that is the conclusion you draw from Maxwell's equations.)

Ring gyroscope

Another application of interference is a a gyroscope, ie. as device to measure rotations.

If the apparatus is rotating, then the pathlengths are different in different directions and so you can use the changes in the interference patterns tomeasure rotations. This is in fact how gyroscopes are implemented in modern aircraft.