The term interference describes wave phenomena originating from a small number of point sources. (By a point source we mean a source of waves having linear dimensions small in comparison to the wavelength of the radiation.) The term diffraction applies to the spreading of waves at apertures and can be thought of as resulting from the interference of many coherent point sources. Sources are coherent when they oscillate with a phase relationship which does not vary in time.

Suppose now instead of a point source we consider an extended source, i.e., a source larger than the wavelength under consideration. Such a source can be considered as a large array of coherent point sources and from our definition of diffraction it should be clear that waves from such a source should exhibit diffraction effects.

In this experiment we wish to gain some feeling for the inherent limitations in optical instruments due to diffraction effects. The light source in this experiment is a helium neon laser. The apertures which we will use are much larger than the wavelength of the laser light.

Figure 1 represents the wave disturbance originating from two point sources, S1 and S2, separated by a distance $d$. By the superposition principle, the intensity observed at the detector will be a maximum (i.e., the waves will be in phase) if the path difference, $\Delta r = r_2-r_1$, is an integral number of wavelengths, i.e. $\Delta r = n\lambda$. But, for a detector far from the sources, $\Delta r = d\sin\theta$. Therefore, the angles at which the intensity will be a maximum, denoted by $\theta_{max}$, obey the following relation:

Figure 1: Interference from two point sources

The intensity observed at the detector will be a minimum if the two sources are out of phase, i.e., $r = \left(n + \frac{1}{2}\right)\lambda$. Therefore, denoting angles where we expect an intensity minimum by $\theta_{min}$, we have

Now, consider an extended source, e.g., a slit, having width $a$ as shown in Fig. 2. We can consider the pattern of diffracted light coming from such an extended source as that resulting from the interference pattern of an infinite number of point sources within the slit, all oscillating in phase.

Figure 2: Single slit diffraction

To find the angles where the intensity will be a minimum, let us divide the extended source of width $a$ into two equal regions: the top and bottom halves of the slit. Now suppose that we are at such an angle $\theta$ that the path difference between a source at the top of the slit and one at the middle of the slit is exactly one-half wavelength; then it is clear that these two will cancel and give no contribution at the detector. Also, it should be clear that the contribution from the entire slit will as well be zero at this angle. This is because for any source in the top half of the slit, we can find one in the bottom half (namely one a distance $a/2$ from the upper source) which will, when added, give zero intensity. Thus, our condition for a minimum intensity becomes

(Note that $n=0$ is not included.) Thus,

The maxima will occur approximately halfway between the minima. A single slit diffraction pattern is shown in Fig. 3.

Figure 3: Single slit diffraction pattern

The pattern of transmission through two slits will depend upon both the width of the slits and their spacing. The amplitude from either slit at angle  is governed by the single slit diffraction pattern - whereas the intensity from the two interfering slits will depend upon the interference pattern from two point sources. An intensity pattern similar to that shown in Fig. 4 will result. Note that the two slit pattern of Fig. 4 would lie entirely beneath the single slit pattern of Fig. 3, showing that the larger scale shape of the two slit pattern is still dictated by the widths of the individual slits.

Figure 4: Two-slit diffraction pattern (with slit separation equal to three times the slit width)

Consider the interference pattern produced by three slits. Slits 1 and 2 together would act as a double slit and therefore produce the first minimum given by Eq. (4). Slits 2 and 3 alone would produce a minimum at exactly the same angle. However, the pattern from the slits 1 and 3 will actually produce a (secondary) maximum at the above angle. The overall amplitude at this angle will be one-third the amplitude from that of the diffraction envelope, because two of the slits cancel each other out. So, the intensity at this angle (a minimum in the two slit pattern) will be one-ninth of the diffraction envelope and will be a maximum.

Fig. 5 depicts the intensity pattern from such a three-slit arrangement. Note that the intensity goes to zero on each side of the secondary maximum.

Figure 5: Three slit diffraction pattern

The point that we wish to stress here is that when we add a third slit to our two slit arrangement, the position of the first minimum moves closer to zero.  As a result, the central maximum becomes narrower. For similar reasons, the maxima in all orders become increasingly narrow as we add more slits. It can be shown that the angular width of the peaks is inversely proportional to N, the number of slits. Thus, the resolution of a slit system, which is its ability to separate two closely spaced wavelengths, is proportional to N.

As noted in the foregoing discussion, the resolution of a slit system increases with increasing number of slits. In the limit as the number of slits gets large, the central maximum of the diffraction pattern becomes narrower. For light normally incident on an equally spaced set of slits (as in Fig. 6), the relation among the diffraction angle $\theta$, the slit separation $d$, and the wavelength of the light $\lambda$ is given by

A grating is most commonly used with a spectrometer to make precise measurements of wavelengths. The spectrometer collimates the incoming light (i.e., makes it parallel) and provides for precise diffraction angle measurements.

In order to determine a wavelength using a grating, one must determine the diffraction order $n$, the grating spacing $d$, and the angle through which the light is diffracted.

Figure 6: Diffraction from a set of slits, showing the relationship among wavelength, slit separation and diffraction angle.

One member of the group should click on the link below to start your group lab notebook. (You may be asked to log into your UChicago Google account if you are not already logged in.) Make sure to share the document with everyone in the group (click the “Share” button in the top right corner of the screen) so each member has access to the notebook after you leave lab.

The diffraction scanner (see Fig 7) is a photodiode (light sensor) attached to a slide potentiometer (a variable resistor). By slowly moving the photodiode across a projected diffraction pattern, one may measure the intensity of light as a function of position. The output of the scanner is sent to the LabQuest Mini (LQM) interface where the intensity and position information is digitized and sent to a computer. Software is used to record and analyze the data. Use of the diffraction scanner can provide precise measurements of diffraction patterns produced by single and double slits (as shown in Fig. 8).

Figure 7: Laser with diffraction scanner

(a)	(b)
Figure 8: (a) single slit with width $a$, (b) double slit with widths $a$ and separation $d$

Place a helium-neon laser and the diffraction scanner on an optical rail, about one meter apart. Place the sliding holder containing the different slits in between the two (but closer to the laser). Note that the slide should have a label saying “this side toward laser”, so make it is oriented correctly.

Turn on the laser and select the 0.08 mm single slit. Project the diffraction pattern onto the diffraction scanner’s screen. Adjust the heights of the laser and scanner so that the diffraction pattern is at the same height as the small hole in the scanner, just above the black tape.

In order to have accurate position measurements given by the slide potentiometer, it is necessary to calibrate it. That is, we must tell the computer the relationship between the voltage from the potentiometer and the actual position of the scanner.

To do so, open the Diffraction Scanner file and complete the following steps:

• Pull down the Experiment menu and select Calibrate.
• Select LabQuest Mini:1 Ch 1: Raw voltage (0-5 V). Click Calibrate Now.
• Move the scanner block all the way to the left limit of its range. Use the millimeter scale mounted below the scanner block to note this position to the nearest half-millimeter. Enter the position, expressed in centimeters (not millimeters) as the “value” for “reading 1.” Click “keep.”
• Move the scanner all the way to the right limit of its range. Use the same edge of the block to note its new position. Enter this final position as the “value” for “reading 2.” Click “keep.”
• Click OK to exit the calibration window.

Next, we need to reset the $x$-axis range to include the calibration limits. To do so, click on the number at the left end of the $x$-axis. Once it is highlighted, enter the desired new minimum value of x. Repeat this process for the maximum value of $x$.

Start collecting data by hitting the COLLECT button. Move the scanner slowly across the pattern and observe the results on the computer screen. You will probably need to practice moving the scanner slowly and smoothly. Try adjusting the gain control on the scanner (the round knob on the top of the device) so that the central maximum of the pattern is almost full-scale on the plot.

Take a screenshot of the intensity vs. position plot for your lab notebook. 

Now try adjusting the gain so that the first-order diffraction peak is full-scale. This setting may be more useful for making detailed measurements.

Take another screenshot of the intensity vs. position plot for your lab notebook at this new scale. 

If there are intensity differences between the right and left sides of the pattern displayed on the computer screen, try very carefully to re-adjust the laser, the diffraction scanner, or the slide containing the slits.

Pull down the Analyze menu and select Examine. A cursor will appear, controlled by the mouse. You may use it to read the positions of the maxima and minima off your diffraction patterns.

Fill in the table provided in Table 1 with your measured values for the positions of minima and subsequent calculations.

Use the Google Colab script below to plot $a\sin\theta$ on the $y$-axis vs. $n$ on the $x$-axis. Perform a fit to the expected shape of your data.

Why do we plot $a\sin\theta$ instead of just $\theta$?

What is the physical significance of the slope of this line? 

From your results, can you determine the the wavelength of the laser (with uncertainties)?

Use the slide having 2, 3, 4, and 5 slits. Observe the diffraction pattern from the double slit.

What is the average separation of the interference maxima?

Move the three-slit opening into the laser beam. The separation of the main peaks should be the same as in the two-slit case – since the slit separation is the same – but a weaker secondary peak should appear.

What happens to the sharpness of the central peak as the number of slits increases? Note that the sharper the main peak the more easily one could separate the main peak from another peak located close to it. The sharpness of the peak is a measure of the resolution of the diffraction system.

Observe the four- and five-slit pattern and qualitatively sketch the patterns. 

How is the resolution (width of the peaks) changing?

Remove the slide containing the slits and look through the narrowest single slit at the single filament lamp (placed in the corner of the lab).

Is the pattern qualitatively different from that you observed as you passed the laser beam through the slit? How is it different?

Look through each of the single slits.

How do the diffraction patterns change as you vary the slit width? Record your observations in your lab notebook.

Figure 9: Grating spectrometer

To explore diffraction gratings, you will use the grating spectrometer illustrated in Fig. 9. A few preliminary adjustments will be needed before accurate wavelength measurements will be possible. For the sake of brevity, we have done some of the alignment procedure for you in advance. Time will not allow the most precise measurements possible with this apparatus!

1. Turn the collimator focus knob until the pencil line on the collimator tube just shows${}^1$.
2. Place the light source at the collimator slit and turn the telescope to look back toward the collimator. While looking through the eyepiece, turn the focusing knob on the telescope (not the collimator) until the collimator slit edges appear sharp. You should see a set of crosshairs. Rotate the eyepiece until the crosshairs are vertical.
3. Adjust the collimator slit to a moderately narrow line. Note that one edge of the slit remains stationary while adjusting the slit width. You will use that stationary edge as the reference point for the crosshairs.
4. Rotate the grating table until the grating appears normal (at right angles to) the collimator. Leave the grating table locked in this position.
5. While looking through the telescope, find the collimator slit and place the intersection point of the crosshairs on the stationary edge of the slit. Read and record the angular position in the Vernier scale window. This position represents the zero-order diffraction. All diffraction angle measurements are to be made relative to this position. The angle and Vernier scales are shown below.

1 Where's the pencil line?

Figure 10: Angle scale reading 20 degrees, 45 minutes

In order to read the angular position, proceed as follows (consulting Fig. 10 as an example):

• Read down from the zero mark on the vernier scale to the next smaller line on the degree scale. (In the example of Fig. 10, the result would be 20.5 degrees or 20 degrees and 30 minutes.)
• Next find the line on the vernier scale which best coincides with a line on the degree scale. (In the example, the best alignment is at 15 minutes.)
• Finally, add the two readings. (In the example, 20 degrees 30 minutes + 15 minutes = 20 degrees 45 minutes.)

Scan the telescope through a large range of angles and observe the spectral lines of mercury. (The spectral lines which you observe are really images of the collimator slit.)

In what sequence do the colors appear as the telescope is moved from smaller to larger diffraction angles?
Note also that the spectrum is repeated as $\theta$ is increased. Use Eq. (5) to explain this repetition.

Change the collimator slit width. You should see the spectral lines change width as you do so.

Move the telescope to find the second order green line. Superimpose the crosshair onto the stationary edge of the green line.

What is the diffraction angle for this line?
Given that the diffraction grating has 600 grooves/mm, use Eq. (5) to determine the wavelength of the green line. Estimate uncertainties.
Is your value consistent with the accepted value of 546.1 nm?

The lab contains the demonstration station illustrated in Fig. 11.

Figure 11: Television diffraction apparatus

The helium-neon laser is equipped with a diverging lens to spread the beam over a wide area to cover the CCD of the TV camera and to reduce the beam intensity. There are several objects provided for the beam to pass through (e.g., a sewing needle, a razor blade edge, and some ink spots on a glass slide.) Move the different objects into the beam to cast a shadow onto the TV camera. Observe the details of these shadows on the TV monitor.

Sketch (or take photos) of the details of each shadow into your lab notebook. Choose two images, and explain (using symmetry or the lack of symmetry) why you see the diffraction pattern that you do.

Make sure to submit your lab notebook by the end of the period. Download a copy of your notebook in PDF format and upload it to the appropriate spot on Canvas. Only one member of the group needs to submit to Canvas, but make sure everyone's name is on the document!

Write your conclusion in a new document (not your lab notebook) and submit that as a PDF to the appropriate assignment on Canvas when you are done. You should write your conclusion by yourself (not in a group), though you are welcome to talk to your group mates to ask questions or to discuss.

The conclusion is your interpretation and discussion of your data. In general, you should ask yourself the following questions as you write (though not every question will be appropriate to every experiment):

• What do your data tell you?
• How do your data match the model (or models) you were comparing against, or to your expectations in general? (Sometimes this means using the $t^{\prime}$ test, but other times it means making qualitative comparisons.)
• Were you able to estimate uncertainties well, or do you see room to make changes or improvements in the technique?
• Do your results lead to new questions?
• Can you think of other ways to extend or improve the experiment?

In about one or two paragraphs, draw conclusions from the data you collected today. Address both the qualitative and quantitative aspects of the experiment and feel free to use plots, tables or anything else from your notebook to support your words. Don't include throw-away statements like “Looks good” or “Agrees pretty well”; instead, try to be precise.