# PHYS 2212 Module 15.3

## Multiple-Slit Interference

15.3 Multiple-Slit Interference

### Learning Objectives

By the end of this section, you will be able to:

• Describe the locations and intensities of secondary maxima for multiple-slit interference

### Diffraction Grating

A diffraction grating is a lot like a pair of slits, only there are more than two slits, more like 1000s of slits in a millimeter.

What makes them particularly useful is the fact that they form a sharper pattern than double slits do. That is, their bright regions are narrower and brighter, while their dark regions are darker.

If instead of two identical slits separated by a distance d there are multiple identical slits, each separated by a distance d, the same effect happens. For example, at all angles θ satisfying dsinθ = mλ we find constructive interference, now from all of the holes (or slits). The difference in the resulting interference pattern lies in those regions that are neither maxima nor minima but rather in between. Here, because more incoming waves are available to interfere, the interference becomes more destructive, making the minima appear broader and the maxima sharper. This explains the appearance of a brilliant array of colors that change as a function of angle when looking at a DVD. A DVD has a large number of small grooves, each reflecting light and becoming a new source like a small slit.

For a given angle, a distinct set of wavelengths will form constructive maxima when the reflected light reaches your eyes. What’s nice is the equations you use for constructive and destructive interference for the double slit also work for the diffraction grating. Constructive interference will occur for wavelengths according to

and destructive interference will occur for wavelengths according to

One application of a diffraction grating is to create an emission spectrum of an excited gas. If you take a gas, say hydrogen H2, and put it in a tube, you can run an electric current through the gas and excite the H2 molecules. When they de-excite, they release light and this light is made from very specific wavelengths. We can’t go into it here (because it is based on quantum mechanics) but every element will emit light with very specific and unique wavelengths.

If we take a lamp filled with H2 gas (called a hydrogen discharge tube) and use a diffraction grating, we can clearly see the wavelengths of the light emitted by the H2 gas.

For hydrogen, these are the specific wavelengths of light that are emitted:

This is called a hydrogen emission spectrum. We can work backwards by measuring the wavelengths of light emitted by an unknown gas, then identify the gas based on those wavelengths. Here are emission spectra for other gases:

All of these emission spectra were obtained by shining the light from a lamp through a diffraction grating.