Interference and Diffraction from Surfaces Molly Golladay Kennesaw Mountain High School

In this lesson, students complete a lab activity that uses a He-Ne laser as a source of coherent visible light. They examine the behavior of the light as it reflects from surfaces. It is expected that they view interference patterns and recognize that the patterns are a result of interference and not a shadow effect. They are also expected to take measurements from an interference pattern and use it to calculate the spacing of a piece of fabric. To perform this calculation, they generate a plot of interference order versus distance from center. The slope of this line will correspond to a constant containing D, the distance from the screen, λ, the wavelength of the laser light, and a, the spacing of the fabric. Since D and λ are known, it is straightforward to calculate a. A magnifying glass will be used to directly measure the spacing so that students may compare their result. They should be able to extend this to explain how one can analyze images and deduce information without being able to see the actual material, and also how scientists use a similar concept to study materials using electron and X-ray diffraction techniques. While this activity is written for an honors-level physics course, it can be adapted for students at all levels.

Problem

How can the wave nature of light be exploited to understand the structure of objects? Objectives

• To understand the effects of light on a surface • To apply this understanding to the subject of materials science

Anticipated Learner Outcomes

After completing this activity, students will be able to: • Understand the difference between interference from a single slit or multiple slits • Identify whether a pattern is the result of interference from a single slit or multiple slits based

on given information • Use light patterns to differentiate between material characteristics • Make the connection between light waves and electron waves and how they are used to study

surfaces Standards NSE Standards

• Content Standard A: As a result of activities in grades 9-12, all students should develop: o Abilities necessary to do scientific inquiry o Understandings about scientific inquiry

• Content Standard B: As a result of activities in grades 9-12, all students should develop an understanding of:

o Structure and properties of matter o Interactions of energy and matter

• Content Standard G: As a result of activities in grades 9-12, all students should develop an understanding of:

o Science as a human endeavor o Nature of scientific knowledge

Georgia Professional Standards

• SCSh1: Students will evaluate the importance of curiosity, honesty, openness, and skepticism in science

• SCSh2: Students will use standard safety practices for all classroom laboratory and field investigations

• SCSh3: Students will identify and investigate problems scientifically • SCSh4: Students will use tools and instruments for observing, measuring, and manipulating

scientific equipment and materials • SCSh5: Students will demonstrate the computation and estimation skills necessary for analyzing

data and developing reasonable scientific explanations • SCSh6: Students will communicate scientific investigations and information clearly • SCSh7: Students will analyze how scientific knowledge is developed • SCSh8: Students will understand important features of the process of scientific inquiry • SP4: Students will analyze the properties and applications of waves

b. Experimentally determine the behavior of waves in various media in terms of reflection, refraction, and diffraction of waves

c. Explain the relationship between the phenomena of interference and the principle of superposition

e. Determine the location and nature of images formed by the reflection or refraction of light • SP6: The student will describe the corrections to Newtonian physics given by quantum mechanics

and relativity when matter is very small, moving fast compared to the speed of light, or very large a. Explain matter as a particle and a wave

Materials/Supplies

• He-Ne laser A laser pointer will work, but the wavelength of light will be different

• Fabric swatches To find suitable fabric, go to a fabric store with a laser pointer and test; ideally, you should have 7-10 different fabrics (2 are too wide, 2 are too narrow, 1 is good enough for data collection, and the remainder have a pattern but it may be indistinct)

• Pieces of phonograph record You may leave the record intact, but pieces are much easier to handle; thrift stores are a good place to look for old LPs

• Cardboard or foamcore Must be rigid enough to create the backing for the screen and the sample holder

• Pushpins • White paper • Office clips

The black clips with a flat side are the best, because they will help the holder and screen stand upright

• Meter stick/ruler • Magnifying glass

To construct the sample holder:

Use a box cutter to cut out a center hole from two pieces of cardboard as shown below. Place a fabric sample in between the two pieces and use pushpins to secure the whole thing:

To construct the screen: Use another piece of cardboard as the backing and clip a piece of white paper on the front. It is easier for students to take data if they just trace the pattern on the paper and then remove it to make their measurements.

The pattern that you should expect to see is shown to the left to make it easier to find appropriate fabric samples. It is the traditional ‘double slit interference’ pattern, only in two dimensions, so it resembles an ‘X’ shape. Due to the restrictions on the size of the openings in the fabric, this ‘X’ will be fairly small (only a few centimeters across even when the screen is ~1m away).

The entire system should be arranged as shown:

Background

This activity explores the phenomenon of multiple-slit interference and its relationship to wave superposition and single-slit diffraction. It is intended to be a springboard to a discussion of modern research techniques that use a similar idea to study surfaces.

Interference should be a very familiar idea to the teacher and can be introduced by a historical discussion. It is relevant to mention that Christian Huygens, in the 1600’s, first proposed the theory of light as a wave, but it was fairly unpopular because Isaac Newton was opposed to the idea. In Newton’s definitive work, Opticks, published in 1706, he advances what he calls a corpuscular theory of light; that is, light as a particle. His main defense of his theory is a discussion of shadows. According to Newton, if light was a wave, shadows would always be diffuse and would not have well-defined edges. But in the noonday sun, it is easy to see that shadows are sharp and have very clean edges. Thus, Newton argues, light must consist of a series of corpuscles (particles) that travel through the air. Shadows are generated by an absence of these corpuscles; the object itself blocks the particles from moving. This theory was logical and supported by the evidence, and so it found wide acceptance, and Huygens’ idea of light waves fell largely by the wayside. However, in 1801, Thomas Young conducted a series of famous experiments that had a curious result. He shined light through a single slit onto a screen on the other side. As expected, he saw one band of light appear on the screen, which fits Newton’s particle theory – the corpuscles that could travel through the slit hit the screen directly in front of the slit, and all other corpuscles were blocked out. But when he shone his light source through two slits instead of one, he did not see merely two bands of light on the screen. Instead, he saw a whole series of alternating light and dark bands, which is in direct conflict with Newton’s theory – the corpuscles shouldn’t be able to get to the parts of the screen that weren’t directly in front of the slits. Later scientists, most importantly Augustin-Jean Fresnel, followed up on Young’s work by observing that for a sufficiently small slit, a pattern of bands could also be produced from just a single slit! Introducing this concept with a historical framework is a good idea because interference and diffraction are conceptually difficult for most students because their intuition regarding light is poor. With a historical introduction, the teacher is presenting ideas that the student will often agree with (e.g., Newton’s shadow explanation for light particles is very convincing) but will then present a

discrepant event, forcing the student to confront their misunderstanding. It also brings a human element to the table and reiterates the fact that science is a human endeavor that can be driven by personality as well as inquiry – if Newton had not been such a dominant and sensitive personality in physics, it is possible that the wave effects of light would have been explored much earlier. This discussion will also pave the way for the introduction of the quantum mechanical concept of wave-particle duality, which resolves the conflict between Newton’s observations and Young’s observations. The effect that Young’s famous Double Slit Experiment illustrates is known as interference. Depending on the order in which the teacher goes through the curriculum, the term may have already arisen in discussions on waves and sound. If this is the case, be sure to remind students of this and what they already know about it. The most important thing to mention is:

Since Young’s experiment demonstrates that light produces an interference pattern, which is an effect limited to waves, then we are able to deduce that light MUST be a wave!

If light were not a wave, an interference pattern could never be produced. The source of the interference pattern comes from a path difference φ. Consider, as shown in Fig 1, a beam of light traveling through two closely spaced slits. To reach the point shown on the screen, the light from the bottom slit has to travel a little bit farther than the light from the top slit – this is known as the path difference.

If the path difference is an integer multiple of the wavelength of the light, then the two waves will arrive at the screen in phase; that is, a crest will coincide with a crest and a trough will coincide with a trough. So the superposition of the waves will be additive, and a bright spot will be seen. This is known as constructive interference. Basically, this is a fancy way of saying, “When we add the two waves together, we get a bigger wave back out.”

If the path difference is a half-integer multiple of the wavelength of the light, then the two waves will arrive at the screen out of phase; a crest will coincide with a trough! The superposition of the waves in this case will be subtractive, and a dark spot will be seen. We call this destructive interference, and what it really means is, “When we add the two waves together, they cancel out.”

If the path difference is something in between, and different parts of the wave overlap. This is a form of destructive interference in that the end result will be dimmer than either of the two waves that are overlapping, but we don’t use this in any of our calculations. It is important here to mention that the slits have to be very close together to observe this effect. As they become farther apart, the pattern becomes less distinct and will eventually disappear. As they become closer, however, the pattern will spread out and become more distinct. The brightness of the far ends will get dimmer, though, so there is an optimum distance between the slits that depends on the wavelength of the light and the distance between the slits and the screen. Ideally, the distance between the slits should be on the order of the wavelength of the incident light, but since we cannot vary ‘slit’ width in our lab, we must adjust the distance between the ‘slits’ and the screen to produce a more optimal pattern. To derive this relationship, we must first do some simple geometry on Fig 1. If we look at the angle θ shown, we can write the path difference in terms of the angle and the distance between the slits, a. Help your students by pointing out the right triangle visible in the figure – the path difference is a leg and a is the hypotenuse. Our relationship looks like the following:

θϕ sina= (1)

Fig 1

It is sufficient to only consider the bright fringes, but you may wish to discuss the dark fringes as well. Recall that for constructive interference, the path difference must be an integer multiple of the wavelength (λ, 2λ, 3λ, etc.), so we can write our equation as:

θλ sinam = (2) where m is the order number, λ is the wavelength of the source, a is the separation between the slits, and θ is the angle shown in Fig 1.

This is often sufficient for most texts, but to complete this lesson, the equation must be further manipulated. The first thing we will take into consideration is the Small Angle Approximation:

θθθ ≈≈ tansin (3)

Depending on your students’ prior math experience, they may have been exposed to this idea before, but it is likely that they have only seen the relationship between the sine of the angle and the angle itself. However, the fact that the tangent is also approximately equal to the sine is the relationship we’re going to use, so make sure that your students have a thorough understanding of this idea. Also, depending on your time constraints, you may wish to have your students actually physically measure the angle (it will be fractions of a degree) and ask them whether or not using the Small Angle Approximation is allowed. By referring back to Fig 1, we can see that the angle indicated on the small right triangle is equal to the angle indicated on the large right triangle. So we can conclude that:

Dd

=θtan (4)

where d is the distance from the center of the screen to the bright spot being measured, and D is the distance from the holder to the screen. Combining equations (2), (3), and (4) gives us the following:

Dadm =λ (5)

Again, many texts would take this as a stopping point, but for this lab, we wish to produce a plot of m vs. d, so it is useful to write m as a function of d.

dDam

=λ

(6)

So the plot of m vs. d should be linear, and its slope will correspond to the constant (a/λD). We use a He-Ne laser because its wavelength is well-known (633 nm) and D is determined by the student, so students will be able to use this slope to determine the spacing of the mesh of the fabric.

Producing this type of pattern with a single slit, as Fresnel and others did, is an effect known as diffraction. While the phenomenon is truly the same as interference, the key difference is in the fact that diffraction is an edge effect resulting from a path difference between a beam of light traveling from the center of the slit and a beam of light traveling from one edge of the slit. The equations are unchanged, except that instead of measuring the distance between the slits, one can determine the width of the slit itself. It should also be emphasized that single-slit diffraction only occurs when the slit width is on the order of the wavelength of the incident light (~0.1 microns, on average). It is critical that students recognize that the phenomenon they are observing in this lab is a result of multiple-slit interference. The holes in the fabric are far too wide to produce single-slit patterns. All of the previous information is commonly included in a high-school physics class, in one form or another, but apart from examining potentially interesting patterns, many students fail to see how this concept can be useful outside of a classroom. The purpose of this lesson is to help students understand that by looking at the images produced by different fabric samples, they can extract information about the surface without ever actually seeing the surface.

This is a concept that modern physicists use very frequently. To study surfaces, materials scientists basically perform the same series of experiments that this lesson outlines, only instead of using light, they use X-rays (in x-ray diffraction), or they use electrons (in electron diffraction). X-ray diffraction is used to look at the bulk of a material, while electron diffraction is used to study the surface of a material. Both of these techniques can help scientists determine information like atomic spacing and structure, surface depth, surface features, and so on. It may be useful to show images resulting from both X-ray and electron diffraction and point out similarities and differences between these images and the lab that students have just completed. Images may be obtained through web search.

Plan Important: It is assumed that all background information except for the ‘real world’ information is covered prior to this lesson. The ‘real world’ information should be discussed AFTER the activity! • Introduce equipment (~10 minutes)

o Students must be careful with the fabric samples so they can be reused o Explain how to set up the system: laser, holder, screen o Emphasize safety procedures for using laser

• Students follow the procedure described in the lab handout (~70 minutes) o Try to stay ‘hands off’ until students are working on collecting data or performing the

analysis; it’s important to keep this activity as exploratory as possible o Make sure that students are placing the holder a good distance (minimum one meter) from

the screen o The handout should be fairly self-explanatory in terms of what the students will actually be

doing • Closure activity (~10 minutes)

o See if there was a ‘best’ sample for the whole class or if different groups picked different fabrics and discuss why it may differ

o Discuss ‘real world’ information in the Background section o Displaying LEED and/or X-ray diffraction images will be very useful

Adaptations

For on-level (conceptual) courses, the slope equation may be given to students, because that is a connection they often struggle with. It might also be useful to adapt the quantitative portion of the activity into a demonstration if students are particularly challenged by the mathematics.

For AP courses, another activity that should also be completed is to use a mercury vapor lamp and a prism to produce several different (coherent) wavelengths of light. Students should view the pattern from their ‘best’ sample using the different wavelengths and observe that as wavelength decreases, the pattern should shrink. This should be incorporated into their concluding remarks. It has also been suggested that a different color laser (e.g., green) could be obtained and students could use their calculated a value to determine the wavelength of the new laser. This result can be checked against the manufacturer’s specifications to again confirm the fabric spacing.

There are many variations on this activity that can be performed, but time constraints may interfere. Assessment/Rubrics

Copies of the student handout and solutions to the conceptual questions follow this document. The handout may be graded in the fashion of your choosing, for completion and correctness on a select number of conceptual questions A few problems on homework/in-class assignments should relate back to the Anticipated Student Outcomes, especially regarding applying this lab to the idea of X-ray or electron diffraction studies.

There should be a minimum of one test question on the summative assessment for this unit that asks about this lab and its application to modern physics in the field of materials science. This can either be an essay-type question or a series of multiple-choice questions. A problem asking students to calculate a spacing given λ, D, and several d values would also be appropriate here.

References Glencoe Science. (2009). Physics: Principles and Problems. Columbus: Glencoe/McGraw-Hill.

Richardson, B. (2006). Physics 102 Study Guide. Unpublished manuscript, Cornell University.

Serway, R.A. (1998). Principles of Physics (2nd ed.). Fort Worth: Saunders CP.

Name: _______________________________________________________________________________

Interference and Diffraction from Surfaces Purpose: 1. To understand the effects of light on a surface 2. To apply this understanding to the subject of materials science Safety Warning: Lasers are very bright and can cause eye damage! Do NOT look directly into a laser beam. Do NOT point the laser beam at anyone. Be careful with the reflections of a laser beam – do NOT look into reflected beams either. Some lasers can also cause skin burns. The laser in this experiment has a low power level so that it should not harm skin, but you should still be careful. Materials: He-Ne laser, piece of record, fabric sample kit, 2 office clips, fabric sample holder, viewing screen, meter stick Part A: Qualitative Exploration 1. Set up your laser and screen as demonstrated. Make sure your laser is not pointed toward anyone

before you turn it on. When not using the laser, turn it off. 2. Reflect the laser beam at a glancing angle from the grooves in the piece of phonograph record so that

the reflected light is shown on the screen. a) Reflect the beam from a spot where the

grooves are running parallel to the beam and sketch what you see on the screen.

b) Reflect the beam from a spot where the grooves are running perpendicular to the beam and sketch what you see on the screen.

c) Describe any differences that you see between (a) and (b). If you saw a pattern that you knew came from a record piece, how could you tell which way the grooves were running?

d) Is this an interference or diffraction effect? Explain your reasoning.

3. In your fabric sample kit, you should have seven labeled samples. Put each sample in the sample holder (one at a time) and hold it in front of the laser so that the laser goes through the sample. For each sample, sketch what you see on the screen in the chart on the right. Make sure to keep the holder at the same distance from the screen for each sample.

4. Could any of these patterns be a shadow reflection of the threads in the

cloth? Explain your reasoning. 5. If any of your samples did not produce a pattern, explain why. 6. If any of your patterns are not a shadow reflection of the threads,

explain why. 7. Which sample number produced the ‘best’ pattern? Explain what you

mean by ‘best.’ 8. Use the sample you chose in #7 for the remainder of this activity. 9. Take your ‘best’ sample and rotate it. Describe what happens to the pattern on the screen as the cloth

rotates. 10. Remove your ‘best’ sample from the holder and stretch it slightly in one direction while holding it in

front of the beam. Describe what happens to the pattern on the screen as the cloth stretches.

Sample Number Pattern

Part B: Quantitative Exploration 11. Put your ‘best’ sample back in the holder. Make sure that your holder and your screen are both

standing straight up and that your laser is completely level. Place your holder a distance D from the screen. Explain why D should be as large as possible.

D = __________ m 12. Once your pattern is distinct and readable, use a pencil to mark the locations of your ‘bright spots.’

You can now remove the paper from the screen and use a ruler to measure the distances of your bright spots from the center spot. You will have two sets of data – record each in the tables below.

Order Distance from Center (m)

-4

-3

-2

-1

1

2

3

4

Order Distance from Center (m)

-4

-3

-2

-1

1

2

3

4

13. Plot m vs. d for both sets of data. Determine the slope of each line and record them below. slope = _____________ slope = _____________ 14. Look in your notes and/or text for a description of interference and diffraction patterns. Recall that we

derived equations for constructive interference and constructive diffraction that are very similar. By looking at these equations, you should be able to determine what the slope of your graph is equal to in terms of the variables: D, λ, and a. Write this relationship below.

15. For a He-Ne laser, λ = 633 nm. Calculate a for both sets of data. Are your values close? Do you

think they should be equal? Why or why not? 16. Using a magnifying glass and a ruler, measure a by counting the number of lines that appear over a set

range (say, 1 cm) and then dividing. Record this value below AND calculate the percent error for your two calculated a values.

17. Given your values for a, is the pattern you are observing an interference effect or a diffraction effect?

Explain your reasoning. Part C: Conclusion/Extension 18. Describe any sources of error in your experiment. Are they significant enough to discard your results?

Why or why not?

19. Given that you now have two methods of determining a, which method do you prefer? Explain your

reasoning. 20. How could you use this experiment to develop a method of classifying fabrics? Is it possible to

determine fabric type by just looking at a pattern? Explain. 21. Scientists use a very similar method to study surfaces of materials – they bounce light off and observe

the patterns from light reflecting through the spacings between the atoms on the surface. Usually, they use electrons (or even X-rays). Explain why this works better than using light.

Solutions to Conceptual Questions 2. c) The patterns are different – the perpendicular pattern should be spaced farther and more

distinct. Because of these differences, one can tell which way the grooves are oriented just by looking at a picture.

d) This is an interference effect – the grooves are spaced too far apart for this to be the result of diffraction.

NOTE: Depending on how you explain interference/diffraction, you may wish to reword this question. For upper-level classes, it should be emphasized that interference and diffraction are the same effect applied to different objects and may therefore not necessarily be distinct terms. In this case, you may want to use ‘multiple-opening interference’ and ‘single-opening interference.’

4. Students should recognize that the patterns that are the exact same size as the laser beam are

producing shadows of the threads – the spacings are too large for interference to occur. 5. Any sample that did not produce a pattern is the result of a tight mesh and/or an opaque

material. Some students may argue that if they turn off the light and bring the screen very close, that they can see something, but it is too indistinct to be sketched. This effect is due to opacity and not to mesh size.

6. They should see that any pattern that’s larger than the laser beam can’t be a mere shadow. 7. ‘Best’ is left to the student to define, but it should approach ‘easiest to see the fringes.’ If a

student is clearly attempting to use a fabric with a poor pattern to perform the next sections, intervene and check their definition of ‘best.’

9. The pattern should rotate as well. 10. The pattern will stretch in the same direction as the pull. 11. D should be large so that the pattern is visible and easy to measure. Again, make sure that

students choose appropriately large distances to get good data 14. The relationship that students should write is that slope = (a/λD) 15. Their two a values should be approximately equal because most fabric has a square weave. 17. This is an interference effect (or ‘multiple-opening interference,’ if you prefer), because a will

clearly be too large to produce single-slit effects. Most fabrics have a weave on the order of 100 microns, which is very large compared to the wavelength of red light.

18. If performed carefully, this experiment should have a very low percent error, which means that

sources of error are minimal. A poorly focused laser, a screen or holder that wasn’t exactly upright, or accidentally moving the screen are really the only sources that could possibly exist, apart from the usual measurement errors from using a ruler/meter stick.

20. Students should recognize that different fabrics produce different patterns and that spacings

can be determined from examining patterns. They could look at a series of images and deduce qualitatively whether the mesh was small, large, or on an interference order. And for interference order images, as long as the screen distance was known, they could easily determine spacing, meaning that different fabrics could be classified according to this system.

21. X-rays and electrons are used because their wavelengths are on the order of atomic spacings, so

interference patterns will be produced. The wavelength of light is too large to produce an interference pattern on such a surface. But apart from that, the idea of developing a classification system is basically the same.