Diffraction and Interference
3
Laser Diffraction
Introduction and Theory
When light waves of a single wavelength pass through a slit and hit a screen, the image is not a single spot of light but a line of spots of varying intensity separated by dark regions. This pattern, the diffraction pattern, occurs because light from one part of the slit interferes with light passing through other parts of the slit. This can produce constructive or destructive interference. If two or more slits are present, the diffraction pattern is complicated by the interference between light passing through the different slits. In this experiment, you will study the diffraction patterns produced by one- and two-slit combinations.
Figure 1. Laser Diffraction Experiment
Figure 1 shows the geometry of the experiment. Consider a double slit experiment in which light of wavelength l is diffracted by two slits each of width a and separated by a distance d.
Of particular interest is the fact that the maxima in the2-slit interference pattern (Fig 2b) are given by
ml = d sinq, m = 0, ± 1, ± 2, …., ± integer. (1)
The minima in the diffraction pattern (Fig 2a) are given by
ml = a sinq, m = ± 1, ± 2, …., ± integer. (2)
P1 If you use one slit instead of 2, only equation (2) applies. Discuss how you could use a single-slit and equation (2) to determine the wavelength of the light source.
In the 2-slit case, the intensity at a point on the screen is actually a diffraction pattern and an interference pattern superimposed on one another. The diffraction pattern is described by the function D(a) = sin2a/a2 which is plotted in Figure 2 (a). This pattern is determined by the width of each slit. The interference pattern is due to the interference of the light arriving at the screen from different slits and is described by cos2d which depends on the distance between the slits. This function is plotted in Figure 2(b). Figure 2(c) shows the resulting intensity pattern when these two functions are combined. This intensity as a function of angle q is given by the equation
, (3)
where I = intensity at angle q,
I0 = intensity at center of pattern,
a = (pa/l)sinq, and
d = (pd/l)sinq.
P2 The first two peaks of the two-slit pattern occur at q=0 and q=sin-1(l/d). Evaluate I for these two values. Are the peaks the same size? Which is brighter?
If a minimum in the diffraction pattern is located at the position of a maximum in the interference pattern, the interference maximum will be eliminated. This is called a "missing order" in the interference pattern. See the points in Fig 2 labeled +l/a.
Procedure
A light sensor with an aperture bracket is mounted on a rotary motion sensor that moves along a linear translator bar.
Set up the software to record signals from the light sensor and rotary motion sensor. The position of the light sensor can be measured as an angle in radians or as a distance along the linear translator bar in cm. Unfortunately, the angle it reads is the angle of rotation of the rotary motion sensor, which is not the angle you need. So be sure to set this up to measure the distance, which you should convert to an angle by trigonometry. Set up to display both a graph and a table of data.
Single-Slit
Position the laser far enough from the light sensor to create a large pattern for good spread in angle, but small enough that you can measure side maxima as well as the central one. Adjust the laser so that it is shining at the midpoint of the aperture opening in front of the light sensor. Use the smallest aperture that will give substantial peaks in your data. First use a single slit with slit width a = 0.04 mm. Measure any distances you will need for your trigonometric calculation. Does the distance from the slit to the laser matter? Why?
Note regarding light sensor: you may need to adjust the setting on the light sensor to get good values – large enough, but not too large. You will then need to divide each intensity value by this setting to get the actual intensity.
Record angle and intensity data while moving the detector by hand along the length of the linear translator. Try to move the motion detector as smoothly as possible. When you examine the spectrum, it should clearly show the large central principal peak and at least two secondary peaks on either side of the center peak. If you don't have these 5 peaks, make additional runs, adjusting positions, alignment, or sensor setting until you do.
Determine the angular position of the minima in this spectrum, and use equation 2 to calculate the wavelength of the laser. Compare to the given wavelength. Be sure you have good agreement before you move on.
P3 What will you do if it doesn’t?
Double-Slit
After you have a suitable spectrum for the single slit, replace it with the double slit for which a = 0.04 mm (the same size as the single-slit) and d = 0.25 mm. Keep the distances the same as they were for the single slit. Repeat the process used to obtain the intensity spectrum of the single slit. Examine the resulting spectrum. It should clearly show a group of principal peaks at the center. Then there will be smaller peaks on either side of the central group, separated from the central group by a region of zero intensity. You should obtain at least seven peaks in your spectrum. You may well have more peaks beyond this minimum number.
Check the position of the regions of zero intensity, corresponding to the minima in the diffraction through each slit (the dotted line in Fig 2). These should occur at the same angles as the minima in the single-slit diffraction pattern. Check that this is the case before moving on.
P4 What will you do if they don’t match?
Graphing
Copy the section of your position and intensity data that is significant (ie, the values are distinguishable from zero).
From the position data, calculate the angle from the central peak – this will be plotted on the x-axis. From the intensity data, calculate the fraction of the maximum intensity – this will be plotted on the y-axis. Since the maximum intensity occurs at the central peak, that point should be (0, 1) or (0,100) if you prefer to do this as percent.
Plot a graph of fraction of maximum intensity vs. angular distance from the center peak.
The theoretical relativity intensity of the peaks is given by the function in equation 3. If you can justify it, you may assume the small-angle approximation, so that
sinq ≈ q
To analyze how closely your data fits this theoretical function, create a new fit function in Origin. Use equation 3 as your fit function with Io, a, and d as fit parameters. Begin with the known values of Io, a, and d. Observe how closely this function fits your data. What is the value of the Chi-squared for this fit? Note whether the fit is better for some peaks than others. Why might this be?
Go through several iterations of fitting (this may take a while), until you are satisfied with your fit. What criteria did you use to decide that you were satisfied? What values of Io, a, and d gave the best fit? What were the errors in these values? Do they agree with your known values within error?
3
Laser Diffraction