Diffraction and Interference 2 The Double Slit, The Diffraction Grating,and Spectroscopy

Page 5-1

Physics 162 Experiment 5 Diffraction and Interference 2

The Double Slit, The Diffraction Grating, and Spectroscopy Equipment: laser pointer, diffraction grating, paper, scissors, tape, pencil, computer screen In Experiment 4, you were introduced to diffraction, waves bending and spreading as they go around an obstacle. You also observed the resulting interference patterns, as the waves combined and either produced larger waves or cancelled each other. For light, constructive interference was shown by the bright patches in the pattern (the waves added), while destructive interference was shown by the dark spots (waves cancelled each other). Using the diffraction pattern, you were able to find the width of a hair. Remember, this pattern made by the laser light hitting the hair is the same pattern as what you’d see if instead of shining the light on a hair, you shone it through a single slit. Part 1: The Double Slit This week we are going to begin by looking at patterns created by a double slit, and then at patterns made by a grating, which has many slits. When a second slit is added, there is an increased amount of interference, and a greater number of dots. Refer to the page of diffraction patterns attached in Appendix A at the end of this lab. They were produced by shining a laser pointer through precisely machined slits. Patterns A and B were made by single slits, while D, E, and F were made by double slits. Compare patterns A and E in Fig. 1 on the next page. Pattern A is produced by a single slit, so it only shows the single-slit diffraction pattern, the dark regions we’ll call fadeouts. Pattern E is produced by a double slit, so it shows both the single-slit fadeouts as well as the bright dots caused by the two slits interfering. (These dots do have dark spots between them, but those are not single-slit fadeouts. They are regions of destructive interference due to light from the two slits interacting.) The arrows on pattern E point to places where the dots are extremely dim or missing in the double slit pattern—these are the fadeouts from single-slit effects. Because the fadeouts for A and E are in about the same place, that means the individual slits must be similar widths. It’s just that E has two of them, so also a double-slit interference pattern.

Page 5-2

Figure 1

Slit A

Slit E

fadeouts So, with the double slit, we have two types of patterns combined, one of which depends on slit width (the fadeouts, from single-slit diffraction). The distance between the two slits is the only other variable, and that distance ends up being responsible for the distance between the dots in the pattern (the two-slit interference). Here is an image of what is taking place when light from two slits interferes:

Figure 2

d

m=1

1

0 θ

Overview Zoom-in on slits

dcenter line center line

}

extra path by lower wave

θ

θ

Page 5-3

Study Figure 2. According to the small triangle on the right, the extra path traveled by the lower wave is related to slit separation “d” and the angle θ by what equation? Hint: “d” is a hypotenuse and the extra path is one side of the right triangle. Extra path = ____________________. Higher order dots occur when the extra path is 2l (m=2), 3l (m=3), etc. That is, the centers of the bright dots in the double slit pattern occur when the extra path (it’s dsinq. Did you get it right?) is a whole number “m” of wavelengths. (We use the Greek letter lambda, l, to mean wavelength.) Hence the brightest spots in the double slit pattern occur when the light from one slit travels at an exact multiple of the wavelength further than from the other slit. Double Slit Maxima: ml = dsinq. Eq. 1 This formula looks the same as what you used last week for the “single slit”; in our case, the hair. Last week our formula was nl = wsinq. The differences are:

1. The variable n refers the number of dark patches you are from the center. In the case of the double-slit pattern, n refers to the number of fadeout you are from the center bright patch. Refer back to figure 1, for pattern E, you have an n=1 fadeout and an n=2 fadeout on each side of the central pattern.

2. The variable w refers to how wide the slit itself is, and the variable d refers to the

distance between double slits. So, both the single slits and double slits have a “w”, a slit width. Only the double slits have a “d”, a distance between the slits. Mixing these two variables is a common mistake, by the way. Take care that you don’t make it.

Activity: Below, you’ll calculate the slit widths and the distance between the slits for the double slit patterns (D, E, and F) attached to this lab. Record your answers on the next page. Assume a distance from the slits to the wall of 1.20 m. You’ll find the angle the same way you did last week. If you can’t print out the patterns, make sure the pdf page width on your screen is 8.5 inches (21.6 cm) before you measure distances. This pattern was made using a green laser pointer with wavelength of 532 nm.

1. What would give you the least uncertainty in doing these calculations (which dots should you use—the ones close to the center or far out)? Explain:

Remember, “m” is the order number of the dot, the center being zero. As always, it is more reliable to measure 2y (the distance from the m on one side to that same m on the other side of the central bright spot) and divide by 2 for the opposite side of your triangle. Look at Figure 2: you’ll use the distance of the center line for the adjacent side of your triangle (in this case, 1.20 meters).

Page 5-4

2. Measure the slit separations. Be careful that dots can sometimes disappear under fadeouts, but you’ll still need to count them to get the right m. Show your work and fill in the following values:

Slit pair D

m = _______ θ = _________ d = ____________ Slit pair E

m = _______ θ = _________ d = ____________ Slit pair F

m = _______ θ = _________ d = ____________

3. Now find the widths of each slit for each of the three patterns. Remember, your m and n values are NOT the same thing, just as d and w aren’t the same. For this, you’ll need to look at the fadeouts. Show your work and fill in the following values:

Slit pair D: w = ___________________ Slit pair E: w = ___________________ Slit pair F: w = ___________________

Page 5-5

Part 2: The Diffraction Grating

In the double slit pattern, places of maximum intensity occur when the extra path traveled by one wave is a whole number of wavelengths, and minima occur when the extra path is .5λ, 1.5λ, 2.5λ, etc. At an angle for which the extra path is 1.1λ, the waves are slightly out of phase, but the intensity will be nearly the maximum value for first order dot (m = 1).

A diffraction grating is an array of slits, and the intensity pattern is a result of the interference of many waves. The detailed pattern for 6 or 8 slits is fairly complex, but when there are hundreds of slits, the effect is clear: the dots are VERY narrow. Here’s why.

Consider diffraction from an array of slits, with waves heading to a point on a screen far away so that the lines of travel are essentially parallel. The path traveled by the one wave is “d sinθ” more than the path of the wave above it.

When this extra path is mλ, each wave will be in phase with the one above it, so all waves will arrive in phase with each other! The grating has a maximum when mλ =d sinθ—the same as the double slit. But the contrast with a double slit is evident when we consider what happens at an angle for which the extra path from one wave to the next is, say, 1.1λ. While one wave is still more or less in phase with the adjacent one, if we look down the grating we see a pattern.

Page 5-6

Compared to the top wave, the second travels 1.1λ more. The third travels 2.2λ more. Skipping down, the sixth wave travels 5.5λ more, and so is exactly out of phase with the top wave, and cancels it. The seventh wave travels 6.6λ more, which puts it at 5.5λ more than the second wave, so they cancel each other. The eighth wave cancels the third, and so on. Thus for d sinθ = 1.1λ, the grating has zero intensity. The contrasts between the grating and the double slit are summarized below.

Page 5-7

Part 3: Determining the wavelength of the laser Now that you understand the physics of the diffraction grating, and the meaning of the diffraction pattern on the screen, it is time to apply that knowledge. Your laser has an indication of its wavelength, but something you should always do in an experiment is calibrate your experimental equipment. Probably that number is not exactly correct. Small defects of fabrication can shift that wavelength a few nanometers from the one indicated. By pointing the laser beam to the diffraction grating and measuring the positions of the bright spots on your screen (as you did before) you should be able to determine that wavelength.

4. Using what you’ve just learned, you need to decide what distance from the diffraction grating to the screen is better. Record your data below, and then calculate the angle you observe:

5. Now consider the equation that relates that distance from grating to screen with the distance between bright spots and the wavelength of the laser (Eq. 1). From this equation you should determine the wavelength of the laser, but first you need to figure out the distance between the lines. Calculate this below, showing your work:

6. Now calculate the wavelength of your laser, showing your work and double-checking your units:

Page 5-8

Part 4: Qualitative spectroscopy You’ve seen that the pattern light creates when passing through a diffraction grating depends on the wavelength of the light. Lasers produce a single wavelength of light (a single color), but most light is a mixture of different colors and therefore made of many wavelengths. White light contains all the colors of the rainbow. Astronomers can use diffraction gratings to determine the chemical composition of stars lightyears away, because different atoms produce different light. Use your diffraction grating to explore three different kinds lights. Different kinds of lights include white LED household lights, colored LEDs (like device power lights), fluorescent lights, incandescent lights, streetlights, neon signs, or the Moon. Don’t use your laser or the Sun. In order to see the diffraction pattern, you’ll have to look at the light through the diffraction grating, holding the grating very close to your eye. The light source will need to appear fairly small, close to a point. Remember that the m=0 peak is in the same place for all wavelengths, so that’s not interesting. You’re looking for the m=1 peaks, where different colors are at different angles.

7. What light did you pick for source 1? Sketch a rough drawing of the spectrum below, labeling colors.

8. What light did you pick for source 2? Sketch a rough drawing of the spectrum below, labeling colors.

9. What light did you pick for source 3? Sketch a rough drawing of the spectrum below, labeling colors.

Now we’ll explore how your computer monitor emits light. Open up the colored lines PDF file, and put it in full-screen mode so it’s the only thing on your screen. (Well, finish reading these directions first, if you’re reading them on your computer.) The room you’re in doesn’t need to be really dark to do this, but darker is better. Turn the monitor brightness up high. The first page is just black. Look through your grating at the screen. If you see anything, it’s from light sources other than your computer monitor, including reflections on the screen or nearby objects. You’ll have to ignore this stuff for the subsequent observations, or make your room darker.

Page 5-9

As you go through the subsequent pages, briefly describe what you observe through the grating, and how it’s different for the different colors.

10. Blue. Describe, and also take a picture of the spectrum with your phone (this takes a little practice).

11. Green. Describe, and also take a picture of the spectrum with your phone.

12. Red. Describe, and also take a picture of the spectrum with your phone.

13. Yellow. Describe. (No picture needed.)

14. Orange. Describe. How is it different from yellow? (No picture needed.)

15. Purple. Describe. (No picture needed.)

16. White. Describe, and also take a picture of the spectrum with your phone.

17. How does your computer screen create colors? When you turn in your lab, upload the four pictures from this section as well.

Page 5-10

Appendix A: Diffraction Patterns for double slits (D, E, and F) and a single slit (B) To produce these images, a green laser of λ = 532 nm was used. Double Slit D

Double Slit E

Double Slit F

Compare the patterns above to the single-slit pattern below. Single slits show only the fadeouts, due to the single-slit diffraction, but do not show the bright dots with dark regions in between (due to double-slit interference). Make sure you can distinguish the fadeouts from the dark spots between bright spots. Single Slit B

v. 2021-02-09