Chapter 17 Knight/Jones/Field Instructor Guide
17
Recommended class days: 3
Wave optics—the interference and diffraction of light—is a continuation of Chapter 16, where interference was introduced. Now, while these ideas are still fresh in students’ minds, we want to extend them to the domain of optics. This is one reason for placing wave optics before ray optics.
Another is the research finding that students have a difficult time distinguishing the domain of wave optics from the domain of ray optics. Teaching wave optics first will allow us to establish a clear criterion for when each is appropriate.
The primary research on student understanding of interference and diffraction is from the
McDermott group (Ambrose et al., 1999). They found that many students have serious difficulties understanding the basic features of the ray model and the wave model of light.
Student explanations of interference and diffraction phenomena tended to be a confused and undifferentiated mixture of features from both models. Even the strongest students in the class had significant conceptual difficulties, and these were found to persist among physics majors in sophomore- and junior-level courses.
In particular, their research has found that:
• After studying wave optics, many students treated all apertures, regardless of width, as narrow slits. These students drew pictures of Huygens waves spreading out from 1-cm-wide apertures. This is a misapplication of the wave model in the domain of ray optics.
• On a post-instruction exam, only 20% of students correctly predicted with correct reasoning that the minima in a single-slit diffraction pattern would move further from the center if the slit were narrowed. Some students misapplied ray-optics reasoning to predict that the minima would move in. Others made a correct prediction, but their reasoning was incorrect and based on incorrect models of light.
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• One group of students employed a hybrid model in which they interpreted the diffraction maximum as being the geometric image of the slit, and they attributed the fringes to “edge effects” of the slit. One student stated, “Light that strikes the edges will be diffracted off.”
• Many students think that no light will pass through a slit if its width a is less than the wavelength

. They state that the light will not “fit” through the slit in this situation. These students did, however, seem to recognize that wavelength is measured along the direction of propagation, perpendicular to the slit dimensions.
• Other students thought that diffraction occurs only if a
 
. One obtains a geometric image of the slit if a
  , but the light “has to bend in order to fit through the slit” when the width is less than the wavelength, and this causes diffraction.
• Students were shown a two-slit interference pattern and asked to predict what would happen if one slit were covered. Only 40% responded correctly. 25% predicted the pattern would be dimmer but unchanged, implying a belief that each slit alone produces the entire pattern.
20% predicted that the right or left half of the pattern would vanish, depending on which slit was covered. Both errors represent ray-optics reasoning in an improper domain.
• Many students think that the standard drawing of a light wave represents an actual spatial extent of the wave. Student diagrams of light passing through a slit show that “part of the amplitude is cut off.” This misconception may be connected to the misconception that electric field vectors extend through space. Students are interpreting diagrams literally, rather than as abstract representations of the situation.
A practical source of trouble for students is the standard figure used to explain double-slit interference. This figure has two difficulties.
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First, it’s rarely explained that we’re seeing the slits “from above,” along the direction of the slits. Without that information, students place a host of misinterpretations on this figure. Second, the figure overlays a graph on a picture of the experiment, with the graph rotated 90° from a
“standard” orientation so that intensity is graphed toward the left. This may be an efficient way to portray the situation, but students need a careful explanation of what is being shown here (and in similar figures later for diffraction), followed by exercises in which they must interpret the figure to answer questions.
In covering the material of this chapter, students will learn to
• Use the wave model of light.
• Understand how the index of refraction affects light waves in a medium.
• Understand how and why interference of light occurs.
• Recognize experimental evidence for the wave nature of light.
• Calculate the interference patterns of double slits and diffraction gratings.
• Understand the conditions for constructive and destructive interference in thin films.
• Understand the inevitable spreading of waves due to diffraction.
• Understand how light diffracts through single slits and circular apertures.
The main theme of this chapter is the inevitable spreading of waves. Nearly all students recognize that water waves spread out after passing through apertures (breaks in a jetty) or around corners
(end of a breakwater). They may not have thought about it, but they’ll readily agree that they can
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Knight/Jones/Field Instructor Guide Chapter 17 hear sound from a source around the corner. You can start this chapter by demonstrating that light also spreads out behind a small aperture. This leads to two fundamental questions:
• What are the consequences of the fact that light is a wave?
• How do we reconcile this observation with the well-known saying, “Light travels in straight lines”?
There is no clear distinction between interference and diffraction ; the usage is primarily historical. In a general sense, diffraction represents the “spreading” of a wave, but the analysis is in terms of the interference of many sources along a wave front. And a “diffraction grating,” which is better thought of as multiple-slit interference, has more in common with double-slit interference than with single-slit diffraction.
Consequently, the order of topics in this chapter is:
• An overview of the various models/aspects of light, with a qualitative look at single-slit diffraction and a quantitative discussion of the speed of light in media. Recognizing that light spreads out behind a slit is a prerequisite for understanding the double-slit experiment.
• An analysis of the double-slit experiment as a special case of the study of interference of spherical waves in Chapter 16.
• An analysis of the diffraction grating as a logical extension of double-slit interference to many slits.
• An analysis of thin-film interference as an extension of the study of interference of two sources along a line in Chapter 16.
• An analysis of single-slit diffraction and brief presentation of the results for circular aperture diffraction.
We’ve found that this approach allows students to gain a more coherent understanding of wave optics, seeing interference and diffraction as important aspects of the wave model of light rather than two distinct and independent behaviors of light.
Sections 17.5 and 17.6, on diffraction, can be treated very briefly if you are pressed for time, particularly if you do not plan on covering the resolution of optical instruments in Section 19.7.
Alternatively, Section 17.4, on thin-film interference, could be skipped without loss of continuity.
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The double slit, the diffraction grating, and the single slit could be covered in two days.
However, this is a difficult chapter for students if you want them to gain any real understanding of what’s going on, rather than blindly using equations for fringe spacings. Three days is much preferred.
DAY 1: Nearly all students know that “light is a wave” and “light travels in a straight line,” but few recognize that these well-known expressions are in conflict with each other. One of the authors likes to start this chapter by firing a laser beam across the room and spraying an aerosol into the beam to show that light travels in a straight line, then sending the beam through a very narrow single slit to produce a diffraction pattern much larger than the diameter of the laser beam.
If you simply ask students whether the diffraction pattern is consistent with light traveling in straight lines, a surprising number will say Yes. The Ambrose paper cited in the Background
Information section found that many students interpret the diffraction maximum as being the geometric image of the slit and the fringes as being due to “edge effects.” To counter this, you’ll want to ask how “bullets of light” would arrive on the screen after passing through a small hole.
Then draw a picture of parallel rays traveling in straight lines through a narrow slit and on to the screen.
Even if there are “edge effects,” which we can’t immediately rule out, you want students to recognize that the geometric image of the slit would be the width of the slit, whereas the central maximum they see on the screen is vastly wider than the slit. That is, they need to recognize that the pattern they see is in conflict with the idea that “light travels in a straight line.”
After recognizing the conflict, analogies with water waves and sound waves that “spread out” after passing through narrow openings suggest that “light is a wave.” Now you’re in a good position to introduce the wave model of light and the ray model of light, noting that this chapter is based on the wave model.
The idea that light spreads out after passing through a narrow slit is essential for understanding the double-slit experiment, because interference fringes are seen only in the region where the two spreading waves overlap. You can easily demonstrate this if you have a single slit and a double slit with the same slit widths.
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With light interference, more than with acoustic interference, students tend to think that the interference exists only on the screen and not in the space between the slits and the screen. A useful demonstration is to orient the slits one above the other, and then use an aerosol spray to illuminate the “fans” of light heading toward the screen. This requires a very dark room, and a green laser tends to be much more visible than a red HeNe laser. If your circumstances are less than ideal, you can use a diffraction grating to make the fingers of the fan brighter and more distinct. You can also show the spatial pattern by sending the light through a tank of water to which a small pinch of corn starch or artificial sweetener has been added. You want students to understand that the bright and dark fringes on the screen are where the antinodal and nodal lines of the interference pattern intersect the screen.
A matched pair of transparencies with concentric circles can be used to show that bright fringes occur where r m

.
Offset their centers (and optionally use a sheet of paper with notches cut into the edge at the locations of the centers to both block out the “backward” waves and to represent the double-slit) and indicate that the circles are the locations of the wave crests from each slit. It is clear where the crests perfectly overlap, and by counting crests at such positions you can show that bright “fans,” just like those in the demonstration described above, are associated with differences in distances to the slits of
A nice double-slit demonstration is to measure the fringe spacing and the distance from the slits to the screen, announce the spacing between the slits (even better, show a microscope image of the double-slit with a scale), and then have the class compute the wavelength of the laser
(633 nm for a HeNe). Rather than using a tape measure, you can pace off the distance to the screen, knowing that each really large step is roughly a meter. Students are amazed that such a crude measurement usually gives the wavelength to within 10%. The remainder of the first day can be spent on example problems of double-slit interference.
Some students apply formulas such as Equations 17.7 and 17.8 without paying attention to the careful (and repeated!) warning in the text that they are small-angle approximations. Other students, who have that warning in mind, can be unsure what do about it in practice: If they don’t know that angle in advance (because it’s not one of the given parameters in a problem), how can they know when the small-angle approximation is valid, and so why should they ever use the approximation if it can get them into trouble? Explicitly pointing out that getting a small angle answer from Equation 17.7 (or y m
L , from Equation 17.8) is enough to ensure the validity of
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Knight/Jones/Field Instructor Guide Chapter 17 those approximations can reassure the latter crowd. Pointing out that getting a large angle answer (larger than 0.1 rad, say) means that you need to go back and use the more exact
Equation 17.6 gives useful guidance to all. Do so in the context of specific numerical examples for maximum impact.
DAY 2: Most textbooks go from double-slit interference to single-slit diffraction and then to diffraction gratings. However, moving from the double-slit experiment to diffraction gratings is a more natural step, since the condition for bright fringes does not change on increasing the number of slits from two. The argument for this presented in the text can be offered in class, but it may be a better use of time to demonstrate two-, three-, and four-slit diffraction (ideally these should have the same slit spacings and widths), and then present a diffraction grating. The increase in sharpness and brightness is obvious on increasing the number of slits.
Some students may struggle with the specification of the slit spacing in terms of “lines per length unit” unless they see an example. It may be useful to start with “lines per inch” to remind students how unit conversion is done, particularly if this is the first chapter you are covering in the second semester of a two-semester course.
Although the maxima of the grating pattern remain located at angles where d sin
 m
 m

, be sure to note that the small-angle approximation is usually not valid for the grating. Some students will try solving diffraction grating problems by using the fringe-spacing result
  
/ for the double slit.
Demonstration: For a quantitative demonstration, if you send a laser beam through a grating and use a meter stick to make reasonably accurate measurements of the screen distance and the distance to the first diffraction spot, you can get a pretty accurate determination of the wavelength. It’s worth commenting on how remarkable it is that you can make reasonably accurate measurements of the wavelength of light with just an ordinary meter stick.
Demonstration: More qualitative demonstrations using gratings can be made by handing out
“rainbow glasses” to the class. Show that the “lenses” are actually gratings by displaying the diffraction pattern of a laser beam passing through one, and then have them look for themselves at a monochromatic point source (a laser spot on a wall is convenient and works well), an incandescent bulb, and finally discrete line sources so that you can make the connection to spectroscopy.
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Thin-film interference should be introduced as being just like the double slit, but the wave sources are not two nearby slits but rather two nearby surfaces which are reflecting the same incident wave. So the question that needs to be addressed is, as always, what is the path length difference? With a sketch like that in Figure 17.19 students can be quickly led to the conditions
2 t
 m
 film
for constructive interference and 2 t
 m
 1
2
 film
for destructive interference—but then you have to give them some bad news and some good news. The bad news is that there’s an extra complication because there are two possible ways that waves can reflect at a boundary; the good news is that the worst that happens is that the roles of the two formulas are swapped.
Emphasize that if they are going to solve any problems they need to establish how many reflective phase changes take place before taking any other steps. It can be helpful to give two examples, a simple one that is a direct application of the constructive or destructive interference conditions (with m

0 or 1 as appropriate), and a “trickier” one where the m values associated with given wavelengths need to be determined.
Demonstration: Unlike two-slit interference, everyone has seen the consequences of thin-film interference. However, there are so many lovely demonstrations it is difficult not to present one or two. Biology students may especially appreciate seeing iridescence. If you can show a Blue
Morpho butterfly mounted in a fully transparent case it is striking to see the wings brilliantly blue in reflected light and a completely different color when illuminated from the other side.
That illustrates a feature of thin-film interference which is not mentioned in the text, namely, when there is destructive interference for reflected light at some wavelength there is constructive interference for the transmitted light, and vice versa. A dichroic filter for stage lighting can be presented in the same manner and then contrasted with a more familiar absorption filter (and the more familiar mechanism of pigmentation for coloration in living creatures).
DAY 3: Diffraction is challenging to explain in quantitative detail. On the first day you probably presented a brief demonstration; today it may be useful to offer more, in the form of a ripple tank
(or video thereof) or a tapered slit, to point out that the narrower the aperture the broader the diffraction pattern. Emphasize that an infinitesimally narrow slit (much narrower than the wavelength) acts just like a point source of light, producing a wave that is the same intensity in every direction; as the slit is widened these waves coming out of every little bit of the slit superpose. The net effect, if you observe the waves at a distance much greater than the width of
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Knight/Jones/Field Instructor Guide Chapter 17 the slit, is the pattern shown in Figure 17.27 with the remarkable formula Equation 17.18.
Because this equation for destructive interference has the same form as the condition for constructive interference for the double-slit it is helpful to highlight the difference. You can then note the small-angle approximations and what happens if the aperture is a circular hole instead of a slit.
There are two conceptual clicker questions that we recommend at this point.
Clicker Question: The first is to present a double-slit interference pattern like that in
Figure Q17.5, making clear which fringe is the center of the pattern, and asking whether this comes from (A) a single slit; (B) a double slit; or (C) there’s no way to tell, it could come from either. Follow-up discussion can focus on why the central fringe in a single slit diffraction pattern is twice as wide as the others.
The second clicker question asks about possible diffraction from two large slits. If the two slits shown in the figure above are (separately, so you don’t worry about “two-slit” effects) illuminated by a broad laser beam, which produces a broader brightly illuminated region on the screen at the right? This sets up a conflict between “common sense” and the equation for the width of the central maximum in single-slit diffraction. Here the equation is inapplicable because the screen is not much farther from the aperture that its width. This question naturally leads to an important numerical exercise: what is the diameter of an aperture for which the diffraction central maximum is the same size as the geometric image of the hole? For
 
500 nm and a screen distance L

1m, the diameter is D

1 mm.
This helps resolve the apparent conflict between “light is a wave” and “light travels in straight lines.” Interference and diffraction effects will just start to show up when apertures sizes are about 1 mm, and they won’t be obvious until the apertures are quite a bit smaller than this. Thus we have a criterion for when the ray model and the wave model of light are valid. Lenses and mirrors are almost always much larger than
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1 mm, so we can use ray optics and not worry about the spreading of the light. The wave model is appropriate when aperture sizes are less than about 1 mm.
If you have sufficient time you may want to outline the derivation of the formula for the locations of the dark fringes in single-slit diffraction. You’ll want students to articulate the fact that the diffraction pattern is due to the entire slit. Some students believe that the central maximum is a geometric image of the slit while the fringes are due to light that “diffracts off” the edges of the slit.
In addition to the specific suggestions made above in the daily lecture outlines, here are some other suggestions for demonstrations, examples, questions, and additional topics that you could weave into your class time.
Squaregrid “gratings.”
The “rainbow glasses” mentioned in the text are typically gratings with square symmetry rather than slits. You can show that a grating with the symmetry of a square produces a square pattern of spots by demonstrating diffraction from a fine wire mesh and from a fine gauze fabric.
Projected soap-film reflection.
This is strongly recommended if you have time, either as a substitute or replacement for the iridescence demonstration mentioned in the Day 2 outline.
Circular aperture diffraction.
This can be helpful to demonstrate (though it is tricky because careful alignment is required) so that students get a sense of scale that is absent in the photo in
Figure 17.28.
1. Any kind of wave spreads out after passing through a small enough gap in a barrier. This phenomenon is known as a. antireflection b. double-slit interference c. refraction
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Knight/Jones/Field Instructor Guide Chapter 17 d. diffraction
2. The wave model of light is needed to explain many of the phenomena discussed in this chapter. Which of the following can be understood without appealing to the wave model? a. single-slit diffraction b. thin-film interference c. sharp-ended shadows d. double-slit interference
A film of oil (index of refraction n oil

1.2
) floats on top of an unknown fluid X (with unknown index of refraction n
X
). The thickness of the oil film is known to be very small, on the order of
10 nm. A beam of white light illuminates the oil from the top, and you observe that there is very little reflected light, much less reflection than at an interface between air and X. What can you say about the index of refraction of X? a. n

X
1.2
b. n
X
 1.2
c. n
X
 1.2
d. There is insufficient information to choose any of the above.
1. Two narrow slits 0.04 mm apart are illuminated by light from a HeNe laser (
 
633 nm ).
What is the angle of the first ( m

1 ) bright fringe?
2. Consider the previous question, but now determine the angle of the thirtieth bright fringe.
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