Interference from Two Sources

Two sources a distance d apart are sending out identical waves in phase. We observe an interference pattern with lines of constructive interference at particular angles and lines of destructive interference at other angles.

When we're at a point far from the sources (far relative to d) then to a good approximation the waves arriving from the two sources are parallel. In this case the path-length difference is d = d sin(θ). We can therefore predict where the lines of constructive interference occur, at angles satisfying:

Condition for constructive interference: d sin(θ) = mλ, where m is any integer.

Similarly the lines of destructive interference occur at angles satisfying:

Condition for destructive interference: d sin(θ) = (m + 1/2) λ

The lines of constructive and destructive interference get further apart if we increase the wavelength or if we move the sources closer together (i.e., reduce d). They get closer together if we do the opposite.

Let's say our two sources are emitting light, and we pick up the interference pattern on a screen a distance L from the slits. When the two sources are in phase there is always a bright spot at the center of the screen due to constructive interference. Other bright spots show up on the screen at distances ym from the center.

tan(θ) = ym/L

Combining this with the fact that the angles for the bright spots are given by the constructive interference equation:

sin(θ) = mλ/d

and that for small angles sin(θ) is approximately equal to tan(θ), the positions of the bright spots occur at:

ym = mλL/d

Similarly the dark spots, from destructive interference, occur at:

y = (m + 1/2) λL/d