2022S 136-3 06 Interference

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Northwestern University *

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136

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Physics

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May 23, 2024

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Lab 6: Wave Interference In this experiment, you will use a virtual simulation of two laser sources to test a model of destructive interference. At the end of this lab, you should be able to describe how to use length measurements to calculate an angle and make those measurements. You should be able to predict the locations of destructive interference and identify locations of both constructive and destructive interference in the data. You will also practice error propagation and Z-score calculation and interpretation. Introduction So far in this course, we have only observed what happens when two waves overlap and interact in one dimension. Our experiment this week will focus on wave interference in two dimensions . We will combine waves from two distinct sources and watch how the interference pattern at different locations in space depends on the relative phase of the waves. Imagine a drop of water falling into a still pond and creating an expanding circle of ripples. Those ripples are a wave – specifically, a transverse travelling wave propagating at a speed determined by gravity and the density and surface tension of the water. Now imagine another drop of water falls elsewhere in the pond, creating a second set of ripples expanding toward the first. When the ripples reach each other, they simply pass through each other. Both waves continue onwards undisturbed. However, the water at the crossing point will move according to the combination of what the two waves demand. Effectively, the water moves according to one ripple and then moves from that height according to the other ripple. So if two peaks collide, the water swells up to form a peak twice as high: a peak stacked on another peak. This is constructive interference . However, if a peak collides with a trough, the water stays still and flat. Effectively, the water wants to form a peak and then form a trough inside that peak, and the net effect is that it doesn’t move at all. This is destructive interference . But in either case, the waves themselves are undisturbed and continue to propagate past the collision point (potentially to interfere again with other ripples). Water ripples are a familiar example, but this description is fully general and applies to any situation where two waves of any sort interfere. In this lab we will be looking at a situation where two synchronized sources create waves which spread out and interfere. Physically, this setup is often accomplished using a water tank, though in our simulation we will use laser light instead. Lasers are different than most ordinary light sources because the light it produces is coherent —each laser source produces a single continuous wave of light, rather than many overlapping similar-but-not-identical waves. This provides a perfect source of waves for our experiment.

Familiarization and Setup First, open the simulation . Click the “Play” button near the center of the screen. We will be using the “Interference” option, using the laser source. a) The source type can be selected in the menu on the right. The laser light option looks like a small laser pointer, as shown at right. b) The green button on each laser source toggles that source on and off. Note that when both sources are on, they are perfectly synchronized. c) Several tools can be dragged out of the upper right panel and used: a ruler, a timer, and an intensity meter. Drop them back in their panel to get rid of them. d) The main panel at right controls the frequency and amplitude of the light sources, as well as the separation between the two sources. We will leave the intensity on maximum to make the effects easiest to see, but we will experiment with varying the other two. Theory We have two wave sources along the left edge of the screen, and we would like to describe the interference pattern that forms on the screen. In other words, we want to know whether any given location will experience destructive interference, constructive interference, or something in between. If we can come up with an equation to predict where each type will occur, we can look at the actual pattern to see if our equation is valid. Our two sources are on the left, separated by a vertical distance d . Let’s follow the wave from each source out in a line θ degrees from the horizontal, as shown in Figure 1. Technically, these are parallel waves which never intersect; however, if d is small compared to the distance travelled, only a small difference in the angles is needed for them to eventually collide. It is a good approximation to claim that these two waves will interfere with each other. The two waves will travel the exact same distance, except that one of them has a slight head start s because of the angle. That head start determines if the two waves are exactly in phase (producing constructive interference), exactly out of phase (producing destructive interference), or in between.

1.1 (10 points): Based on the diagram above, use geometry and trigonometry to find a formula for s in terms of d and θ . Write your equation in the box below. Show your work or provide a quick explanation of how you arrived at the equation. If this “head start” distance s is exactly equal to one wavelength, then we will get perfect constructive interference. More generally, this happens if s is equal to any integer m number of wavelengths. We will call m the order of the interference. So constructive interference occurs along spokes radiating from between the two sources, at angles satisfying: λm = d sin ( θ ) (1) By the convention we’ve been using, θ is the angle above the horizontal. But note that negative-order interferences also exist (because m can be negative), which just means that θ will also be negative. So negative orders are just those interferences which occur below the horizontal. We can make a similar argument for destructive interference. If s is a ½ wavelength (or 1 ½, 2 ½, etc) then the waves will be exactly out of sync. So destructive interference occurs at: λ ( m + 1 / 2 )= d sin ( θ ) (2) “head start” Figure 1: two identical waves from separated sources are leading in the same direction. We are interested if they will reinforce or cancel (or something in between).If this amount is half a wavelength, then the waves are exactly OUT of sync and we get destructive interference. If this amount is exactly one full wavelength, then the waves are exactly IN sync and we get constructive interference s = d sin ( θ )

Observing Interference The conditions of interest are the source separation d and the wavelength λ . The source separation can be set directly by the simulation controls. Set the source separation d to be 1500nm (the default value). 2.1 (5 points): We cannot control the wavelength directly, but we can control the frequency and then measure the wavelength. Set the frequency to its default value at the center of the frequency range. (If you’ve changed the frequency already, you can reset to default using the orange button in the lower right corner. Be sure to re-select the laser mode and source separation too.) Now measure the wavelength λ and record it in the box below. To do this, turn one of the lasers on and the other off. Run the simulation until the waves have filled the entire screen and stabilized, then pause. The wavelength is the distance between two adjacent crests or troughs, but we can get more precision by measuring the distance between two distant crests/troughs with the ruler and dividing by the number of waves between them. Show your work and don’t forget units! Now, turn both lasers on. If our hypothesis is correct, we should see some angles where the waves undergo destructive interference and thus have very low amplitudes. The intensity of these radiating spokes should be halfway between a peak (bright color) and a trough (black), so they will look dim and greyish. The angle these lines make with the horizontal is our angle θ from Figure 1 and Equation 2. They are numbered by order : the first line above horizontal is m = 0 (so that m + 1 / 2 is positive), the next above that is m = 1 , the first line below horizontal is m =− 1 , etc. (Note: in theory we should try to verify both Equations 1 and 2. However, the destructive interference is much easier to see, so we will test only Equation 2.) Ideally, we should measure the angle θ for each line of destructive interference. Unfortunately, the simulation neglected to supply a protractor. But really, to verify Equation 2 we don’t need to measure θ itself, we just need to measure sin ( θ ) . We can do this using trigonometry. If we hy potenuse opposite side adjacent side Figure 2: Example diagram for finding via trigonometry. distance = 2930.6 nm , ¿ of crests = 6 λ = 2930.6 nm 6 = 488 nm

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