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PC2232 Physics for Electrical Engineers
Single Slit Diffraction
Objectives
- To investigate the diffraction patterns produced by monochromatic laser light for various slit configurations..
- To determine the wavelength of the laser light from measurements of the intensity distribution of light for the single slit.
- To determine the thickness of a human hair strand from the diffraction pattern.
Equipment
- Single slit wheel with holder, aperture disk, diode laser, multiple slit wheel with holder
- Computer, interface device, Data Studio software
- Light sensor, motion sensor, precision linear translator, optical bench
- Micrometer screw gauage
Theory
When coherent light of monochromatic wavelength is incident upon a slit, the light diverges as it passes through the slit in a process known as diffraction. A laser produces coherent light, which means all the light striking the slit is in phase. If the light then falls on a screen placed at a large distance from the slit, it produces a pattern of alternating bright and dark images of the slit. This pattern is referred to as a Fraunhofer diffraction pattern, which is the simplest case of diffraction. It occurs when rays emerging from the slit can be considered to be parallel. The diffraction process is explained by the fact that light is a form of electromagnetic wave and the different portions of the slit behave as if they were separate sources of light waves – Huygens’ principle. At each point on the screen, the light from the different portions of the slit will have a different phase due to the different path length of light from each portion of the slit to the point on the screen. Light from the different portions of the slit will interfere with each other and the resultant intensity will vary at different places on the screen. This is illustrated in Figure 1 where five rays are shown from different portions of the slit. The choice of five is arbitrary and the slit could be divided into any number of portions.
Figure 1: Path of five rays from different portions of the slit.
Consider wave 1 from near the bottom of the slit and wave 3 from the center of the slit. They differ in path length by the amount (a sin θ)/2 as shown, where a is the width of the slit and θ is the angle each of the rays makes with the horizontal. It is also true that the path difference between rays 2 and 4 is (a sin θ)/2. If this path difference is exactly equal to half a wavelength, i.e., λ/2, the two waves will be 180◦^ out of phase and will interfere destructively and cancel each other. Therefore, waves from the upper half of the slit will be 180◦^ out of phase with waves from the lower half of the slit when
a 2
sin θ =
λ 2
By dividing the slits into 2m portions, it can be shown that the condition for destruc- tive interference will be satisfied at angles θ on the screen above and below the center of the pattern given by
sin θ = m
λ a
(m = ± 1 , ± 2 , ± 3 ,.. .) (2)
There is no simple expression for the location of the maxima on the screen other than that for the principal maximum at the central of the pattern. The other maxima are much less intense than the principal maximum and are located approximately halfway between the minima. The diffraction pattern that appears on a screen will have an intensity variation above and below the slit as shown in Figure 2.
Figure 2: Diffraction pattern on a screen distance L from a single slit with width a.
Figure 4: Diffraction of light as a function of the ratio λ/a.
To understand the generic features on the intensity variation (4) without having to make a particular choice for the slit width a, it is convenient to rewrite this expression in terms of the quantity α:
α ≡
a sin θ λ
Then, equation (4) becomes
I(θ) = I 0
[ sin(πα) πα
] 2 (6)
In the above expression, the argument of the sine function is in radians.
Figure 5: Diffraction of light as a function of the ratio λ/a.
The plot of equation (6) is shown in Figure 5. It can be seen that the diffraction pattern formed when light wave passes through a rectangular-shaped slit consists of a set of bright spots (the principal maximum and many secondary maxima) interspersed with regions of darkness (the minima). A summary of the key features of this rectangular-slit diffraction pattern is as follows:
- Minima. The minima (locations of zero light intensity) occur at the angle θ given by α = a sin θ/λ = ± 1 , ± 2 , ± 3 , ± 4... and are called first, second, third, fourth, ... minima, respectively. It is to be noted that the condition for minima is the same as obtained in (2) from a simpler geometrical argument.
- Principal Maxima. The central peak, bracketed by the two “first minima” (which are located at α = a sin θ/λ = ±1), is the region of highest light intensity and most of the diffracted wave’s energy is concentrated in this region.
- Secondary Maxima. A detailed analysis of equation (6) (which involves taking derivatives to find the maxima of this expression) reveals that the secondary maxima occur at angles θ given by α = a sin θ/λ = ± 1. 43030 , ± 2. 45902 , ± 3. 47089 , ± 4. 47741 and are called the first, second, third and fourth secondary maxima are only 4.7%, 1 .6%, 0.8%, 0.5% respectively of I 0 (the intensity of the principal maximum).
Figure 8: Setting up the equipment.
to the light intensity. At the side of the light sensor, there is three buttons which select the electronic amplification factor. Set the gain to appropriate levels for different parts your experiment (usually 0 – 100 for obtaining the full pattern without “clipping”, and 0 – 1 for measuring the intensity of secondary maxima).
Figure 9: Light sensor and aperture bracket.
Note: Since the detector area of silicon pho- todiode is fairly large (about 8 mm in diam- eter), an aperture disk is placed in front of the sensor to restrict the part of the light pattern that would hit the photodiode. The aperture disk has several slit apertures la- beled 1 through 6 with the following widths: 1 = 0 .1 mm, 2 = 0 .2 mm, 3 = 0 .3 mm, 4 = 0.5 mm, 5 = 1.0 mm and 6 = 1.5 mm.
- Prepare the computer interface and software settings
- Connect the Powerlink interface to the computer, turn on the interface and the computer.
- Connect the light sensor and rotatory motion sensor to the interface if neces- sary.
- Launch Data Studio.
- Select the Rack and Pinion option in the Setup section for the rotatory motion sensor to measure the Linear Position of the sensor directly.
- The default rate of measurement (data acquisition) is 10 Hz. Set this to a higher rate if necessary.
- Reset the graph to choose Intensity for the y-axis, and linear position for the x-axis.
Part A: Single Slit Diffraction Pattern (Qualitative)
In this part of the experiment, you will observe qualitatively the single slit diffraction pattern due to a wide and narrow slits respectively.
A-1. Rotate the single slit wheel until the laser light beam is incident on the variable slit. By rotating the disk, the laser light passes through a slit whose width can be continuously varied from a = 0.20 mm to a = 0.02 mm. Position the disk so that the light is passing through the widest portion of the variable slit where a = 0.20 mm. Use the adjustment screws on the back of the diode laser to adjust the vertical position of the beam if necessary. Rotate the single slit disk so that the laser passes through a narrower and narrower portion of the variable slit. Observe the diffraction pattern on the white screen of the aperture bracket (see Figure 10). Note the spatial width and overall brightness of the central peak as the slit becomes narrower.
Figure 10: Diffraction pattern on the white screen.
A-2. Identify the central peak, which is the brightest region of light at the center of the pattern and is bracketed by two dark spots (the first minima). Also note the much dimmer secondary maxima symmetrically located on both sides of the central maxima and separated from each other by the second, third, fourth,... minima.
Part C: Diffraction Pattern of A Human Hair
Diffraction of light may be used to determine the thickness of a human hair. When a coherent beam of light is shone on a single strand of hair, the beam diffracts around the edges of the hair and casts an diffraction pattern. The interference pattern cast by hair shares some similar features with that of a narrow single slit. The same equation describing diffraction minima for the single slit can be used to describe the diffraction minima for a strand of human hair. In this part of the experiment, the thickness of your hair strand will be estimated by examining the positions of the minima of its diffraction pattern.
C-1. Pull or cut a strand of hair from your head and mount it vertically and directly in front of the laser beam. C-2. Adjust the position of the hair or the laser beam if necessary so that the beam is centered on the strand of hair. C-3. Determine the distance from the strand of hair to the screen and record the hair- to-screen distance as L in Data Table 4. C-4. Obtain a diffraction pattern using similar procedure as in Part B. Concentrate on the minima of the pattern. C-5. Locate first minima (two of them appear symmetrically to the right and left of the central peak) on the graph. Use “Smart Tool” to measure the distance for each first minima from the central peak respectively and record them in Data Table 4. C-6. Repeat the previous step for the second, third, fourth and fifth minima (if still visible). C-7. Print the graph and label each each secondary maxima and minima on both sides of the central peak. Title this graph “Single Slit Diffraction Pattern (Human Hair Strand)”. C-8. Tape about 2 cm of the hair used for the experiment to the graph.. C-9. Measure and record the thickness of the hair strand as d in Data Table 4. You may use the mircrometer screw gauge provided or any other method that you think can give better results. Describe briefly your method if you do not use the micrometer screw gauge.
After you have finished taking all the necessary measurements, you may want to spend some time examining the diffraction/interference patterns of the other configurations, in- cluding those on the multiple slit wheel. Your report will not include this – it is meant to complement what you have learnt and seen in the lectures.
