# The Proof of Light as a Wave

Light is a wave! OK, now that we have established that, let's recall one of
the characteristics of waves that we have previously studied: interference. When two or more waves in a medium interact, they create interferences between the waves. You may recall this from our
experiments with the ripple tanks or simply from watching the wave patterns from tossing a couple of rocks in the water next to each other. When the waves interact such that crests and troughs
reinforce each other, we have *constructive interference*. When they interact such that crests and troughs cancel each other, we have *destructive interference*.

**Constructive
Interference - The waves mutually
reinforce each other to produce a larger wave.**

**Destructive
Interference - The waves act to
cancel each other out at the points where they meet.**

There is also an infinite combination when waves meet where there is some interference but it is neither totally constructive or destructive. In 1801, Thomas Young performed an experiment which demonstrated a remarkable property of light. Young shined light through two tiny slits in a piece of cardboard and observed, not two bands of light as might be expected, but several alternating bands of light and dark. This became some of the most compelling evidence to date of the wave nature of light, and the experiment became known as "Young's Double Slit Experiment".

Here's what was happening: As the light passed through the two small slits, the waves passing through interfered with each other much as the ripples in a pond do when a pair of rocks is tossed in. The bands of light on the screen showed where the waves were interfering with constructive interference at that point, while the dark areas showed where the destructive interference was taking place.

This brings us to a term called *Path Length Difference (PLD).* PLD is
the difference in distances traveled by two waves. If two waves with the same wavelength leave a point in phase, they will remain in phase as they travel. If

they both reach the same point at the same time, they will still be in phase and interference will be constructive.

But if the path one wave travels is longer or shorter than the path the
other wave travels, then when they meet, they may or may not be in phase. This is where Path Length Difference comes in. If the PLD (the difference between the two distances traveled) equals one
wavelength, then the waves will be in phase and will constructively interfere. If the PLD is exactly one half of a wavelength (**l**), then there will be total destructive interference. The bands of light and dark in a
Double Slit Experiment can be explained in terms of the number of path length differences that the waves travel. When the waves from both slits travel the same distance towards the center, we get a
bright center band and PLD is 0. On either side of the midpoint are alternating bands of light and dark. The light bands correspond to the points where the PLD is 1**l**, 2 **l**, 3 **l**, etc for the two waves. The dark bands correspond to the points where the PLD
is **l**/2, 3**l**/2, 5**l**/2, etc for the two waves. The formula for determining where the light bands will fall is given by:

where m is the *order* or number of bands from the center,
**l** is the wavelength of the incident light, d is the distance between the two slits,
and **q** is the angle measured between the centerline and the appropriate order of the interference
fringe on the screen. Similarly, we can predict the location of
the dark fringes (mutual destructive interference) with the formula

where m + 1/2 indicates that the path length difference is an additional half a wavelength and the wave cancel each other. Theoretically there is an infinite number of orders that may be seen, but in practice, we generally see just a few.

There is a symmetry that allows us to actually measure these distances and compute the angles and wavelength of the light and this is an experiment we will perform in class. Consider a double slit experiment as shown below:

If we know the distance from the slits to the screen (generally on the
order of a meter) and we measure the distance between fringes (on the order of 10^{-3} m) , we can use trig to determine the angle **q** as shown above. Our PLD will be the wavelength (or the order times the wavelength for
bright fringes beyond the first fringe). The angle we found above is, by similar triangles, the same angle shown below.

Usually d, the distance between slits, will be on the order of
10^{-4} meters. Thus, using our formula

ml=dsinq

we are able to determine the wavelength of the light.

To read what others have to say on the subject, try:

http://www.glenbrook.k12.il.us/gbssci/phys/Class/light/u12l1a.html

http://www.glenbrook.k12.il.us/gbssci/phys/Class/light/u12l1b.html

http://www.glenbrook.k12.il.us/gbssci/phys/Class/light/u12l3c.html

http://www.glenbrook.k12.il.us/gbssci/phys/Class/light/u12l3d.html

The NTNU Virtual Physics Laboratory provides several excellent applets that demonstrate principles of Physics. Click Here for an applet you can run from the NTNU Virtual Physics Laboratory that will show single and double slit interference.

For Practice Problems, Try: *Giancoli Multiple Choice Practice Questions (Questions
1-12)*