Diffraction and Young s Single Slit Experiment


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1 Diffraction and Young s Single Slit Experiment Developers AB Overby Objectives Preparation Background The objectives of this experiment are to observe Fraunhofer, or farfield, diffraction through a single slit and to use this observation to determine the width of the single slit. Read sections regarding wave phenomena, especially diffraction, in textbook. Also, read texts concerning opamps and the laser safety document in the reference section. This experiment involves working with lasers so it is important to understand the dangers of eye exposure to laser radiation. You will be using a Class 2 laser in the visible spectrum of light. DO NOT stare directly into the beam, permanent or temporary eye damage can and will occur. With that said, a quick flash will not damage the eye. The body s natural blink reflex will allow the eye enough time to cool and prevent damage. It is safe to view the reflection of the beam off of a wall or screen as well. The experimental procedure will involve wave diffraction. Since light travels as an electromagnetic wave, it will be used in this experiment. Diffraction can be described as the bending of waves around objects and the divergence of waves passing through small openings. The HuygensFresnel principle explains how this is possible by reasoning that every point of an advancing wave is itself a source of new waves. Following this principle we can see that if a wave were to meet a small gap in an obstacle that new wave sources would be created in this gap and would propagate forward from that position. Figure 1 shows how the HuygensFresnel principle can describe diffraction in this manner. For diffraction to occur the gap, or obstacle, the wave is hitting should have a size on the order of the wavelength of the wave. If it is too large then diffraction will not be observed; think of shining a light through a doorway, it just continues along its path. Now, when the gap has a size on order of the wavelength the new wave sources form a new wavefront on the other side of the gap. This new wavefront is determined by the summing of the phases and amplitudes of each wave source. This summing describes constructive and destructive interferences of the wave sources and the interferences create points of maxima and minima along the wavefront. As the new wavefront propagates outward it becomes planar at very long distances. The two most likely types of diffraction you will deal with are Fresnel and Fraunhofer diffraction. Fresnel, or nearfield, diffraction deals with distances from the aperture where the wavefront is not yet planar. Fresnel diffraction will change both the size and shape of the observed wave as distance from the diffraction aperture changes. Fraunhofer, or farfield, diffraction deals with distances far enough from the aperture where the wavefront has become planar. Fraunhofer diffraction changes only the size of the observed wave as the distance from the aperture changes.
2 Figure 1: HuygensFresnel Principle This experiment will deal with farfield diffraction. To determine whether we are dealing with nearfield or farfield diffraction we use the Fresnel number equation from figure 2. F being the Fresnel number and a being the size of the aperture or the slit width in our case. L is the distance from the diffraction aperture to the observing screen and λ is the wavelength of the diffracted wave. Fresnel diffraction is observed at F 1, while Fraunhofer, or farfield, diffraction will occur when F << 1. For values much higher than 1 we would use optics geometry. Assuming we know the wavelength and the slit width we can adjust the distance to the screen to ensure farfield diffraction. Figure 2: Fresnel Number Equation Now knowing that we can ensure that we are dealing with farfield diffraction and knowing that the wavefront we will observe is expected to have minima and maxima we should learn how to determine the minima and maxima locations. Since the edges of the gap will determine where destructive interference occurs we can find expected minima locations with trigonometry as shown in figure 3. Marking D as the distance from aperture to screen, a as the slit width, and y as the location of a minima we can find the location of the first minima by using the equation below. Minima occur when sin(θ ) = m*λ/a where m 1 and describes 1 st, 2 nd, and higher minima. a sin(θ ) = m*λ
3 Figure 3: Determining Minima of Wavefront However, as the distance from the aperture increases ϴ approaches ϴ and small angle approximation can be used. The equation above becomes or even simpler, y can be determined by a sin(θ) = m*λ There is less than 1% error in the small angle approximation out to ϴ = 14. So as long as sufficient distance is between the aperture and observing screen the simplified equation will work. After finding the wavefront s minima, it is possible to calculate the wavefront s intensity as a function of the location along the observing screen. The original wave arrives at the slit, or gap, as a planar wave made up many point sources. Each point source uniformly differs by a constant phase displacement. You can imagine these sources summing by adding tiny arrows tiptotail with their direction determined by their phase displacement. The actual calculation of these sums is beyond the scope of this lab, but for more information there are several Richard Feynman lectures on QED that you can look up. The measured intensity is determined by these relative intensities and depends on the total phase displacement. The intensity as a function of y can be found by the equation below.
4 Where λ a, y, D, and λ hold the same values as in the small angle approximation. The output of this equation should result in an intensity plot similar to figure 4. Figure 4: Intensity Plot The laser you will be using is rated at <1mW. This laser s specifications, including wavelength, can be found in the data sheet listed in the reference section. To power the laser you will need to supply 3V to the Vcc marked pin on the data sheet. The best way to do so is with a voltage follower. The output of the voltage follower should feed directly to the laser. Consult your DC circuit fundamentals book if unsure how to build a voltage follower. References (Type A Laser Package) Materials The equipment and components required to perform this experiment are:
5 ANDY Board 1 ea VCSEL (APCD C2A) 1 ea Mount for VCSEL 2 ea Aperture for Diffraction Lens 1 ea LF356N OpAmp Procedure Analysis: 21. Using MATLAB model the intensity function. The program should take wavelength in nm and slit width and distance to screen in meters. The y in the intensity equation above will be your xaxis. Use the value 1 for I o and be sure to limit the x axis to only 3 or 4 minima locations on each side. (You can use MATLAB to calculate minima locations and then scale the x axis accordingly). 22. Assuming a slit width of 200 μm and using the Fresnel number equation, calculate the distance the observing screen should be from the aperture for Fraunhofer, or farfield, diffraction to occur. Measurements: 23. Position the laser with aperture #1 attached on a flat stable surface and facing a smooth wall about 2 meters away. Ensure that the laser is facing perpendicular to the wall. (Avoid any angle from the observed wave to the point source.) Write down the distance to observing screen. 24. Supply power to the system, and determine the distance to the first minima on the observing screen. (For more accuracy measure the distance between the 2 first minima on either side of the center and divide in half.) Write down the distance to the first minima. 25. Using the small angle approximation, determine the slit width of the aperture. (All information is available, laser information can be found in the data sheet.) Write down the slit width of the aperture. 26. Using the MATLAB program above create an intensity plot for the aperture. Include at least 3 minima on either side of the maxima. Save an image of this plot. 27. Does the MATLAB plot match the observed image on the screen? 28. Repeat steps 3 6 for aperture #2.
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