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Interference - Flathead Valley Community College Flipbook PDF
12 CHAPTER 3. INTERFERENCE Figure 3.1: Interference geometry from a double slit, Knight, Physics 3.2.1 Interference Doub
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Chapter 3
Interference 3.1
Objective
Observe and measure the interference produced when laser light is incident on single and double slits. Use the measurements to determine wavelength of the laser.
3.2
Theory
The double slit experiment done by Thomas Young has become one of the most often repeated labs in Physics. That is because it is the litmus test for waves. Only quantities with wave properties will interfere when they pass through a double slit. We will use laser light incident on a slit pair to produce interference. The laser light has the unique property of coherence, which produces a single color, non-varying phase and a constant polarization between the two slits making it easy to get the waves to interfere. Thus the double slit experiment validates the idea that light is a wave. Incredibly, this experiment has also been used to validate that electrons also act like waves. Interference of waves produce constructive and destructive areas on a screen behind the illuminated slits. The pattern is affected by the width of the slits and the distance or separation between them. In fact, interference is also produced by a single slit. This phenomenon is generally refer to as single slit diffraction. Our goal in this laboratory is to produce, observe and measure the interference patterns produced by slits. We will: (1) observe the different fringe patterns produced by interference and diffraction, (2) observe and measure the effects of slit-width and slit spacing on fringe patterns, and (3) experimentally measure the wavelength of the Helium-Neon laser. Since we have 2 effects in this experiment, we will describe each in turn and establish distinct variables for each effect. 11
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CHAPTER 3. INTERFERENCE
Figure 3.1: Interference geometry from a double slit, Knight, Physics
3.2.1
Interference
Double-slit Interference: The general picture for double-slit interference is given in Fig.3.1. This is an annotated figure from our textbook by Knight. The equations for maximum and minimum are given as: dyn nλ = d sin θ = (3.1) L 1 dyn (n + )λ = d sin θ = 2 L
(3.2)
These equations can be solved for yn which is the distance on the screen from the central maximum to the interference order we are measuring. The order is given by the integer n. In Fig.3.1 the 2nd order constructive fringe is shown with the y2 maximum fringe being
3.2. THEORY
13
highlighted. So, by measuring the distance y on the screen to a particular maximum or minimum, and knowing what the slit separation d is, we can easily solve for the wavelength of the light.
3.2.2
Diffraction
The general picture for single slit diffraction is given in Fig.3.2. Again, the figure is from Knight’s Physics text. The equation for finding the minimums is: pλ = a sin θ =
awp 2L
(3.3)
Here the integer p is used to represent the order for diffraction minimums. The diffraction maximums are more difficult to determine and we do not need them in this experiment. Note that we measure width, w between paired minimums rather that using y. The purpose is simply to avoid trying to estimate where the center of the central maximum might be, a potential source of errors in measurement. The consequence of using w is that a factor of 2 appears in the denominator compared with the formulas in most textbooks. As with the double slits, equation 3.3 can be solved for λ thus giving another method for determining the wavelength of the helium-neon laser used in this experiment. The actual patterns you will see on the screen in this lab will be a combination of two slit interference and diffraction. This is simple due to the fact that the slits in a 2-slit mask have a finite width which produces diffraction. The diffraction pattern will act like an overall intensity envelope on the 2-slit interference pattern. In Fig. 3.1 the diffraction pattern is present only to the extent that we see the fringes inside of a central diffraction maximum. If the image extended wider we would see a minimum in the pattern and then the first order diffraction lobes.
3.2.3
Counting Fringes
In practice, to find the order of the fringes, (p for single slit minimums and n for double slits) we count the number of fringes in a given distance on the screen. We will use the variable N as the count. The trick is to relate the count N to the order needed for our equations. The concept is slightly different for the double slit pattern compared to the single slit. When you see the interference pattern for a double slit, you will have a hard time determining where the central maximum is located. Consequently, it is easier to count the number of interference fringes in an arbitrary interval. The idea is to pick two dark fringes, measure the distance between them, then count the number of bright fringes in that interval. In Figure 3.3 shows the interval between the 1st and 4th dark fringes. The count N of bright fringes is 3. The width is ∆y then related to the fringe count by equation 3.4.
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CHAPTER 3. INTERFERENCE
Figure 3.2: Interference geometry from a single slit, Knight, Physics
3.3. PROCEDURE
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Figure 3.3: Counting bright fringes in between dark fringes. Knight, Physics
λ=
d∆y NL
(3.4)
The process for counting fringes in the single slit is different because it is easy to identify the central maximum. But if we count the central maximum we have to realize that it is twice as wide as the separation between adjacent minimums. Studying the x-axis in Figure 3.4 shows this. Since the fringe count between minimums is really the number of equal intervals in a given distance, the count must be N + 1 if the central maximum is included. Subsequently the formula for wavelength when counting fringes in the diffraction pattern is given by equation 3.5. The central maximum is so much stronger than the adjacent maximums, so it is wise to count fringes including the central. λ=
3.3
wa (N + 1)L
(3.5)
Procedure
The basic experiment set up is shown in Figure 3.5. The equipment list is outlined and followed with the procedure.
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CHAPTER 3. INTERFERENCE
Figure 3.4: Counting bright fringes in between dark fringes, but count central maximum as 2. Knight, Physics
1 2 3 4 5 1
2 3 4 5 6 7
He-Ne Laser Optical Rail Slit mask set and mounts Screen with measured rules on it Tape measure to measure distance slit to screen
Set up the laser, slit mask and screen with the 4 cm rule markings on it. The distance between the slits and the screen is L. This distance can be set at any value that is convenient. Measurements can be simplified if you get the interference patter on the screen, then move the screen along the rail until minimums are on the two outside hash marks on the screen. This will force the distance in the interval to be 4 cm which is the ∆y value. Measure 5 different intervals recording the fringe count N, L, ∆y and slit separation d. Average the data and determine the wavelength of the He-Ne laser. Repeat the procedure for slits with a different distance between them. Change to the single slit pattern and get a diffraction pattern on the screen. Move the screen until minimums are conveniently on the outside hash marks of the screen. This will again force the distance w to be 4 cm. Count the number of bright fringes and calculate the wavelength. Repeat the measurement for 2 other slit widths.
3.4. QUESTIONS
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Figure 3.5: Experimental arrangement for measuring interference.
8
Compare the calculated values or λ to the accepted value of 632.8 nm. Accuracy% = 100 ∗
3.4
λtrue − λmeas λtrue
(3.6)
Questions
For full Credit on your report, be sure to answer all of the following questions.
1 2 3 4
Derive the formula for wavelength measured with a single slit pattern where the central maximum is not included in the measurement. Describe a missing order. If the double slits are separated by greater and greater distances, what happens to the interference fringe pattern? The 4th maximum in an interference pattern is found to be missing. What must be the relationship between the width of the slits and the separation of the slits?