For your report: In addition to the usual heading and goals, include in your report those items indicated below. Note that Part A requires that an electronic prelab be completed first, and Part C requires that a Word file be submitted first.

 Goals

1. to measure the wavelength of laser light using the Young's double slit method
2. to use the measured wavelength to measure the track spacing on a CD
3. to compare the interference patterns obtained from single, double, and multiple slits and to interpret the results obtained

 Equipment

Collect the following equipment.

• Single, double, and multiple slits on a piece of transparency film  [These are provided in your lab kit. If you misplaced the slits that were given to you, open this file in a photo-editing program like Paint or Adobe Photoshop. Print the file first on paper to make sure it prints correctly. Use at least a 600 dpi printer. The size of the image on the paper should only be 1" x 2". If it's printing larger, that's because your application is changing the resolution of the image. (Web browsers will do this.) Be sure to print from an application that maintains the image resolution. Photo-editing programs will generally do this. Once you've figured out how to get the image to print at 1" x 2", print it on transparency film.]

• Laser pointer (red)

• Meter stick

• Tape measure (optional)

• CD (blank or burnt)

• Ring stand and clamp (to hold CD): You may be able to borrow this from your science department. If not, use your ingenuity to find a way to hold the CD in Part B. One possibility is to hold it between the pages of a book which is, in turn, clamped shut with a rubber band.

• Sticky tape

It's best if you can do the experiment in a very dark room. Having someone assist you may also be necessary.

Identification of the slits:  The slits are numbered on the transparency film.  Here's information about them.

Here's how the measurements given above were made. First, a file with a resolution of 600 pixels/inch was created. Narrow white lines were drawn on a black background. All of the lines for the double and multiple slits were 1 pixel wide. That works out to (1/600 in)(25.4 mm/inch) = 0.0423 mm.  The single slits were 3 and 5 pixels wide. Separations for the double and multiple slits ranged from 2-4 pixels. Separation were measured from the center of one slit to the center of the next.

 Part A.  Wavelength of Laser Light

Prelab

1. Read sections 1 and 2 of Chapter 28.
2. Study the theory of the double-slit experiment below.
3. Do the related online exercises.

Theory of the Double-Slit Experiment

First, let's review the concept of path difference as it relates to interference of waves. This was introduced in Chapter 14 in the study of sound waves. See P209. The same considerations apply to light. If two sources of periodic waves are separated in space, then waves will travel different distances to any point away from the sources. The waves will interfere constructively at that point if the difference in the distances traveled by the waves to that point is an integral multiple of the wavelength. The waves will interfere destructively at that point if the difference in the distances traveled by the waves to that point is an odd half-integral multiple of the wavelength.

For an illustration, open and play this applet. Waves of the same speed, frequency, wavelength, and phase pass through two closely-spaced apertures on the left. While actual waves spread out in different directions beyond the apertures, the applet only displays two waves at a time. These are colored red and blue to help distinguish them. The two waves shown trace out a path to the same point P on the screen. For the particular point chosen, note that the waves arrive at the point in phase. That is, their peaks coincide in space. Therefore, point P is a point of constructive interference. If the waves were sound waves, an intensification of sound would be heard at that point. If the waves were light waves, the intensity of light would reach a maximum at that point. In order to see the positions of constructive interference along the screen, enter 1 for Show fringes.  Points of destructive interference occur midway between the red fringes. Enter 2.0 cm for the vertical position on the screen, reset, and play the animation. Where do the waves meet this time? What kind of interference occurs at point P?

In order to see the effects that the wavelength, slit separation, and horizontal distance to the screen have on the fringe positions, try changing each of these in turn. There is a simple relationship between these parameters. We'll investigate that next.

Open and play this applet. This presents the same situation as the previous applet; however, this time the light waves are represented by straight lines for convenience. Additional lines have been drawn for the purpose of investigating geometrical relationships. Several points have been marked for easy reference. The area near the slits is shown in the enlarged view. Both views are shown below.

Full view	Enlarged view near slits

We make these definitions:

• PS₁ and PS₂ are the distances from the sources to point P on the screen.
• CD is the perpendicular bisector of the line S₁S₂ joining the slits. We'll name these distances L = CD and d = S₁S₂.
• QS₁ is drawn so that PQ = PS₁. This makes QS₂ the path difference. We'll call the path difference z = QS₂ for short.
• The distance PD is the perpendicular distance from line CD to the point of interest on the screen. We'll call this distance y.
• We'll also define angle θ as the angle PCD and angle α as the angle S₂S₁Q.

The proof below will involve some approximations that are typically made for such experiments. The approximations work extremely well because of the distance scales involved in a typical double-slit experiment. The screen is far from the slits compared to the separation of the slits.

1. First note that line CP is approximately perpendicular to S₁Q. As a result, we can say that angle α is approximately equal to angle θ. That's because the two angles have corresponding sides perpendicular or nearly so.
2. Triangle S₂S₁Q is a close approximation to a right triangle. Therefore, sinα is very nearly equal to z/d. Since α is nearly equal to angle θ, we have this relationship: sinθ = z/d.
3. That's the geometry. Now we use the physics of interfering waves. z, the path difference, is equal to mλ when the interference is constructive. Therefore, the condition for constructive interference for the double-slit experiment is:

mλ =  dsinθ, where m is an integer.

4. This is the same condition that you used in P223. We'll take this a step further and note that sinθ is approximately equal to y/L. We can say that because angle θ is so small that the hypotenuse CP of the triangle PCD is very nearly equal to leg L. (Note that in the double-slit experiment which you will do, L will be more than 10 times greater than y.) Therefore, mλ is nearly equal to dy/L, and the wavelength is given by:

λ = dy/(mL).          [Equation 1]

This relationship verifies the results that you may have noted earlier, namely, that for a given m, the position of the bright fringe on the screen increases as λ and L increase and decreases as d increases. That is, y = mLλ/d.

Now do the WebAssign assessment L19A. Afterwards, continue below.

Method

In recording measurements, identify quantities with the same symbols as are used in introduction above.

The method is fundamentally like that of Young's experiment described in section 28-2. You have the advantage, though, of being able to use a laser and digitally-created slits. An overhead view of the experimental arrangement is shown below. You'll need to work in a room that can be darkened, since the interference fringes are faint.

Here are some design considerations:

• If you hold the double slits vertical, the pattern of fringes on the wall will be spread horizontally.  If you hold the slits horizontal, the fringes will be spread vertically. 
• You'll need a way to hold both the double slit and the laser steady while taking measurements. Perhaps you can tape the slits to the front of the laser. If you do so, be careful not to cover the slits themselves with tape.
• The slits needs to be 4 to 5 m from a wall in order to provide distance for the interference pattern to spread out. This will make the fringes wider, and you can measure them more accurately.
• Tape a sheet of typing or copy paper to the wall in the area where the interference pattern appears.

Now here's what you do.

1. Use the double slits marked 1 (d = 0.127 mm).

2. Turn off the lights and shine the laser through the slits in order to cast the interference pattern on the paper taped to the wall. Be sure to align the laser with the slits. One way that works is to hold the slits vertical and then slowly scan the laser horizontally, sweeping through the double slit as you do.  You're looking for a pattern of faint red fringes on the wall  It will look something like the photo below. The fringes will be about 1-2 cm apart.

3. With a pencil, trace the outline of the fringes onto the paper.  Also, clearly mark the positions of the center of the central bright fringe and the centers of the two dark fringes on either side of the central bright fringe. Beside the tracing, write the slit separation in order to distinguish it from the next tracing that you'll do.

4. Shift the paper up or down a few inches to provide a clear area for your next tracing. Using the double slits marked 2 (d = 0.169 mm), repeat steps 2 and 3.

5. Measure the distance from double slit to wall to the nearest centimeter. Record this distance on your tracing.

Analysis

Include items 1-8 below as well as your tracings in your report.

Now you're ready to take measurements from your tracings.  Do the following for each of the two tracings.

1. Label the maxima (bright fringes) m = 0, m = ±1, etc. You'll need to judge where the midpoints of the fringes are. See the example tracing below.

2. Measure the distances y to the nearest 0.1 cm from the m = 0 line to each of the maxima at m = -1 and +1. Record your results in a table like the following. Note the units of measurement.

3. For each value of y, calculate the wavelength in units of nanometers using Equation 1. Then calculate the mean wavelength, the deviations in the wavelength, the mean deviation, and the percentage mean deviation.

4. Give the wavelength range written on the label of your laser pointer.

5. Using the center of the wavelength range given in step 4 as the accepted value, calculate the experimental error between that value and your mean measured wavelength.

6. Describe your methods for a) positioning the laser and the double slits and holding them steady, b) for measuring the distance from slits to screen, and c) for measuring distances on your tracing.

7. Does the percentage mean deviation of your measurements account for the experimental error? Tell why or why not.

8. Qualitatively discuss potential sources of error that would contribute to the deviations in your measured values of wavelength.

 Part B. Track spacing of a CD

Prelab:  Read sections 4 and 6 of Chapter 28.

Include items 1-6 in your report.

The tracks of a CD are very closely spaced and can be used as a reflection diffraction grating. (See section 28-6.) Now that you know the wavelength of the light from your laser, you can use that information to measure the CD track spacing. Do the following to set up for the measurement.

1. Arrange the CD and laser as in the diagrams below. Both side and overhead views are shown. If you don't have a ring stand, tape the CD to some kind of vertical support such as a book stood on end. You can hold the laser by hand, but it will be easier to keep the interference pattern steady if you can set the laser on something. Aim the laser at the CD's surface. Position the laser so that it doesn't block the interference pattern. A wall will serve as your screen. Note that unlike Part A where the slits were 4-5 m from the wall, the CD need only be a meter from the wall. You'll know why when you see the diffraction pattern.

Side view	Overhead view

2. Remembering that your goal is to measure the track spacing on the CD, take all the measurements you need in order to achieve that goal. Record your measurements. Describe clearly in words the interference pattern that you saw. Compare and contrast it to the pattern that you observed in Part A.

3. Clearly describe how you made your measurements in step 2. Your descriptions must leave no doubt in the reader's mind what you measured and how you measured.

4. Starting with the appropriate formula, calculate the track spacing of the CD. Use the value of wavelength that you measured in Part A.

5. Do a search on the internet for the value of the track spacing on a typical CD. State the value and give the URL of the source.

6. Calculate the experimental error between your calculated value of the track spacing and the value you found in your search.

 Part C. Comparative observations

Prelab:  Part C requires sections 4 and 6 of Chapter 28.

You'll need to download an applet for this part. Right-click here and save the file ejs_diffraction.jar to your hard drive. Be sure that the file saves with the extension jar. When you double click on the filename on your hard drive, the applet should open.

This exercise will help you gain familiarity with the characteristics of single-, double-, and multiple-slit interference. Open the applet that you downloaded. Note the various controls for the number of slits, slit width, slit separation, wavelength, intensity, and resolution. See the diagram to the right for the distinction between slit width and separation. For the following exercises, keep the resolution constant at the maximum value. You shouldn't need to change the intensity, but if you do, be sure to keep the peaks of the intensity curve visible. Note that two representations are provided. The fringes at the bottom of the screen are similar to what you would actually see in an experiment. The intensity curve is a graphical representation of light intensity as a function of position. In order to see the fringes only, you can click off the intensity.

For the following, the term fringe spacing represents the distance between the centers of the bright fringes. Fringe width is the width of a bright fringe. Fringe sharpness is the ratio of the width of a bright fringe to the dark space between fringes. The smaller this ratio is--that is, the smaller the bright space compared to the dark space--the sharper the fringes are.  Relative intensity refers to relative heights of the intensity peaks. As you make the changes described below, observe how the fringe spacing, width, and sharpness and the relative intensity change.

1. Set the slit width to 1 µm. While keeping the wavelength constant, change the number of slits from 1 to 20.
2. For the same slit width and N = 2 , investigate the effect of changing the slit separation.
3. For N = 1, investigate the effect of changing the slit width. Then, for a given slit width, change N to 2 to see how that affects the width of the central peak.

Download this Word template and use it for recording your answers to parts a-f.

1. The slits that you'll use later for observations are those listed in the table below. Write the condition--in equation form--for constructive or destructive interference for double, multiple, and single slits. Look these up in the text if you need to. The formulas must be in terms of wavelength, λ, slit separation d (or slit width W in the case of N = 1), angle θ (as defined in the text), and order (m).  In addition to writing the equation, indicate the values of m (integer, odd half-integer) for which the formula applies and indicate whether the condition is for constructive or destructive interference.

Now answer the following. Identify the slits by the ID given in the table. For N > 2 (multiple), assume that the slits have the same width. You needn't give explanations at this point.

2. For N > 1, which slits do you expect to produce the same fringe spacing? _________

3. For N > 1, which slits do you expect to produce the greatest fringe spacing? __________

4. Which slits do you expect to produce the sharpest fringes? _________

5. For N = 1, which slit do you expect to have the broader central intensity peak? __________

6. For N = 1, how do you expect the width of the central peak to compare to the widths of the other peaks in the intensity curve?

Before continuing below, submit your Word file.

Method and Analysis

Include items 1-3 in your report.

1. Position yourself 4-5 m from a light-colored wall as you did in Part A. You can tape white paper to the wall in order to have a plain, white background. Shine the laser through each of the 6 sets of slits in turn. Examine the interference patterns from all of the slits before writing anything. Then examine each pattern once again. For each of items b-f in the prelab, decide whether your predictions agree with your observations. If you find discrepancies, revisit your observations as well as your predictions in order to determine where the problem lies.

2. Click on this link to open a page with three sets of axes. Print the page. On the axes, sketch graphs of intensity as a function of position for slits 1, 4, and 6. We pick these because they provide for comparisons for single, double, and multiple slits of the same width or separation. Draw the graphs with the same position scale for visual comparison of fringe spacing and sharpness. If, for example, two of the sets of slits have maxima of the same order at the same position, then your graphs must show that. If one set of slits has fringes much narrower or brighter than another set, your graphs must show that.  A position of 0 represents the position directly opposite the slits on the screen. This would, for example, be the center of the central maximum for a double slit pattern. Use the dashed lines to help line up features of the graphs that appear at the same positions. Draw neatly and make the largest peaks use the full intensity scale. Make sure that distinguishing features of the graphs are clearly visible. Use your textbook and the applet as needed for help in drawing the graphs.

3. Based on the theory of single-, double-, and multiple-slit interference, explain your predictions for each of items b-f of the prelab. If you determined that a prediction was incorrect, base your explanation on a correct prediction. List your explanations and label them b to f for clarity.

 Conclusion

Review what you did in this lab and provide a well-organized conclusion summarizing what you did and what you learned. Organize it into three paragraphs, one for each of Parts A - C.

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