Diffraction at Circular Edges

Transcription

1 Diffraction at Circular Edges References: Equipment: Born and Wolf, Principles of Optics nd ed., Pergamon Press 1964 pp Giles, Robin A., Waves and Optics Simulations:The Consortium for Upper Level Physics Software,pp Pasco Optics System Manual (OS-9141) Experiment 8, pp a) Red laser (λ = 63.8 nm), Green laser (λ = nm) b) Pasco Photometer (model 800) with optic probe, Linear translator, Light Source Apertures, Fresnel Zone Plates, Photometer apertures, Optical bench, Lenses (-mm, 18mm, 48mm), Standard component carriers, Science workshop, Capstone computer software, Canon Optura 40 Video Camera, mage J and Virtual Dub Video Software. ntroduction: This experiment studies the behavior of light that passes through circular apertures of small diameter. With the appropriate selection of aperture size, the pattern produced on a viewing screen is not just the bright disk that a simple ray treatment would lead you to expect. nstead, the screen shows a bright central disk, the Airy disk, and a pattern of alternate dark and light rings concentric with the central disk. This is a diffraction pattern of a relatively simple geometry. Diffraction patterns occur in general when the light arriving at a given point has traveled there from the source by different paths. f light is considered a wave, the difference in path length brings about a difference in phase, and so the possibility of constructive or destructive interference at the point of arrival. This leads to alternate light and dark regions that are, in the present case, circular due to the geometry of the apparatus. Note that similar experiments can also be done with microwave apparatus (eg. See Cenco Microwave Optics Demonstration Apparatus Manual pp 13-16),you might also want try to do some of those experiments. The interference effects would be washed out to a great extent if the light source generated many different frequencies, as an incandescent bulb does. So this experiment uses its light source a laser, which generates light of a single fixed frequency

2 Before doing any experiments with the laser, read the description of it with a great deal of care. Pay particular attention to the safety precautions. This is a very safe laser designed for student use, but no laser light should ever be looked at, directly or highly reflected, since all the light energy it carries is concentrated in a beam of very small cross sectional area. This experimental write-up gives supplemental information, theory, and tips for the Pasco Optics System Manual experiment 8 (DFFRACTON AT CRCULAR EDGES). Please follow the experimental instructions listed there. Also please be patient with any experiment with optics, these types of experiments often require careful alignment, leveling, and tweaking to achieve good results. Note that if the laser has not been warmed up for at least 1.5 hours there may be fluctuations in the beam intensity due to the expanding glass tube. t is highly recommended that you do these experiments in a darkened room. Several methods are described below to measure diffraction at circular edges; each method has strengths and weaknesses. Choose the particular experimental parts that interest you and which you think you can finish in your assigned lab sessions. Theory: When a plane wave impinges on a circular aperture of radius r, diffraction theory predicts (using symmetry and some approximations) a diffraction pattern that consists of a bright central disk (Airy disk) concentrically surrounded by bright and dark rings of constructive and destructive interference (similar to a bull s eye pattern). The intensity distribution is theoretically predicted to be: πr sinθ J1 θ λ = πr sinθ 0 λ where! is the intensity of the central bright spot J! is the Bessel function of the first kind r is the radius of the aperture θ is the angle value calculated using trig λ is the wavelength of the laser n this experiment you will verify aspects of this intensity distribution

3 Procedure: Airy Disk Photometer Method 1. General set-up a. Use a fiber optic cable to connect a linear translator to a photometer b. Place the linear translator on a small lab jack just past the end of an optics bench c. Attach a slide containing a circular aperture (Pasco 9165-D) to a component carrier and position it on the optics bench 40 cm from the tip of the fiber optic cable inside the linear translator d. Place a laser about 5 cm behind the aperture and direct it through the aperture at the linear translator e. Use the lab jack to position the linear translator so that the laser beam is directly incident upon the fiber optic cable. Photometer set-up a. Turn the VARABLE knob all the way clockwise b. Set the SENSTVTY knob to 10 (the needle should go off the end of the scale; this is ok) c. Cover the front of the linear translator with your hand to block all light and use the ZERO ADJUST knob to move the needle exactly to 0 d. Uncover the linear translator and use the knob on its side to precisely find the point of maximum intensity e. At the point of maximum intensity, use the VARABLE knob to set the photometer needle to 10 i. The scale is now set so that the photometer measures intensities relative to the central maximum. A reading of 10 on the scale corresponds to 100% relative intensity while a reading of corresponds to 0% relative intensity, etc. f. Use the linear translator to measure the distance from the central 0 th order maximum to all the maxima and minima you can locate. t may be difficult to measure any max/minima beyond the first order i. To measure the distance between two points with the linear translator, simply count how many turns of the knob it takes to move from one point to the other. The numbers on the knob correspond to 0.1 mm so a full turn of the knob corresponds to 1 mm. DO NOT try to use the scale on the bottom of the inside of the linear translator. t is not as accurate

4 ii. The 0 th order minimum is the point at which the needle stops decreasing right before it begins to increase. t won t always register as 0 intensity on the photometer due to ambient light in the room. iii. Record the distances between the center and the maxima/minima and record the relative intensities at all the maxima. Airy disk interference patters are governed by the following equation: θ =!!! ( 1.) where θ is the angle in radians from the center of the pattern to a feature (a maximum or minimum), λ is the wavelength of light being used d is the diameter of the aperture. m is a constant that depends on the order of the feature you re measuring and whether it is a maximum or minimum. The following table lists values for m: TABLE 1 Order Value of m for maximum Value of m for minimum 0 th st nd rd th Use basic trigonometry and your measured distances between the center and each of the max/minima to determine the angles in radians at which the maxima and minima occur. Compare these experimental values to the theoretical ones obtained using the above equation. f you have time/interest take data using the green laser (wavelength = nm)

5 Recall that the equation that governs the intensities of each of the maxima is: πr sinθ J1 θ λ = πr sinθ 0 λ (.) thus 0 J1( x) = x where πr sinθ x = (3.) λ J1( x) The first three minima of = are at x = , , and the first four 0 x maximum occur at x = 0.000, 5.135, 8.417, and Can you show this analytically or numerically? The first four maxima of and J1( x) = x are then calculated to be = 1.000, , 0.004, 0 Compare your experimentally measured maximum relative intensities with the theoretical J1( x) maximum values of = listed above. Note you may be only able to reliably measure 0 x the relative intensity of the second maximum. Bessel functions are typically studied in more advanced physics and mathematics courses but you can easily plot them on your own using Maple (available on all lab computers). To calculate the output of the Bessel function, simply enter BesselJ(1, x); evalf(%);.below is a Maple graph of the Bessel function involved:

6 J1 ( x) Exercise: Plot x using Maple or Mathematica. Compare the graph with your experimental data of. Do they agree qualitatively and/or quantitatively? Also compare your 0 sin x data with a Maple/Mathematica graph of x diffraction from a single slit (recall that for a single slit = ( + cos ) Photometer with Motorized Linear Translator Method which is the theoretical relative intensity for 0 sin x 1 θ ). x 1. General set-up a. Use a fiber optic cable to connect a linear translator to a photometer b. Use the output jacks on the back of the photometer to connect it to the Science Workshop 750 nterface c. Place the linear translator on a small lab jack just past the end of an optics bench d. Turn on the linear translator and allow it to move all the way to one side e. Attach a slide containing a circular aperture (Pasco 9165-D) to a component carrier and position it on the optics bench 40 cm from the tip of the fiber optic cable inside the linear translator f. Place a laser about 5 cm behind the aperture and direct it through the aperture at the linear translator

7 g. Use the lab jack to position the linear translator so that the laser beam is directly incident upon the fiber optic cable. On your computer, open Capstone a. Click Create Experiment b. On the image of the Science Workshop nterface, click the input to which your photometer is currently connected i. Select Light Sensor ii. Set the Sample Rate to 5 iii. Close the Experiment Setup window 3. Collect data a. Turn on the linear translator in the opposite direction as before b. Click the Start button in Capstone to begin collecting data c. Once the linear translator has just about finished traveling to the other side, turn it off d. Click the Stop button in Capstone to stop collecting data 4. n the Displays sidebar, click and drag Graph into the gray space in the window to show a graph of the data. Capstone will present a graph of voltage vs. time. This information alone isn t very useful for analyzing the data. However, voltage is proportional to intensity. That is, if you flip the voltage graph 180, it effectively becomes a graph of intensity. Use the graph to determine the difference in intensity between the central peak and the lower peaks. You will have to zoom in very close on the base of the central peak in order to see the smaller peaks. The minimum intensity of the graph is most likely not zero so when calculating the intensity of a peak, use the difference in intensity from the absolute minimum to the peak. Compare your experimental results to the theoretical values

8 Sample plot with first order maxima visible on either side of central peak To calculate the angular position of a maximum, use the speed of the linear translator (10 mm/min) to convert time to distance. Then use this calculated distance between maxima, the known distance from the aperture to the tip of the fiber optic cable and some basic trigonometry to calculate the angular position of the maxima. Compare your experimental results to the theoretical values. Video Camera Method (Please check for availability of video camera) 1. General set-up a. Place each end of an optics bench on lab jacks b. Place a Canon Optura on is optical mount as close to the edge of the optics bench as possible c. Position a laser on the optics bench so that its front end is about 40 cm from the video camera lens d. Attach a slide with a small circular aperture to a component carrier and position it about 6 cm from the front end of the laser. The 0.08 mm diameter aperture on slide 9165-D in the Pasco Advanced Optics kit works best i. Make sure the laser beam passes directly through the aperture ii. Put black electrical tape over the other apertures and diffraction gratings on the slide to prevent unwanted interference

9 . Camera set-up a. Remove the camera s lens cap, connect it to a computer via FireWire and power it on b. Put the camera in night mode i. Press the NGHT MODE button on the left side of the camera. A small moon symbol should appear in the top left corner of the camera screen c. Set the camera for infinity focus i. Make sure the camera is first set for autofocus. f manual focus is on, MF will appear on the left side of the camera screen. Pressing the FOCUS/DATA CODE button will toggle autofocus on and off ii. Press and hold the FOCUS/DATA CODE button until MF appears on the left side of the camera screen d. Zoom in as far as possible on the laser beam passing through the aperture 1. Adjust the camera s height and move the camera around to find a good image of the Airy disk. t should look something like this: e. Adjust the exposure i. f the central bright spot of the Airy disk is saturating the image, manually adjust the camera s exposure 1. Press EXP on the back of the camera. Use the scroll wheel underneath the EXP button to decrease the exposure 3. Make sure not to decrease the exposure to the point that the outer bright rings disappear

10 3. Data capture a. Open VirtualDub b. File > Capture AV c. Device > 1 Microsoft DV Camera and VCR (DirectShow) d. File > Set capture file i. Specify where to save the video e. To begin, Capture > Capture video i. A clip of only a second or two is sufficient f. To stop the capture, Capture > Stop capture g. File > Exit capture mode h. File > Open video file i. Open the clip i. File > Save as AV i. Resave the clip as an AV using a different file name from before j. Analyze the clip with magej i. Open mage J ii. File > Open 1. Open the AV clip iii. Press OK on the AV Reader dialog box that appears iv. mage > Stacks > Z Project 1. n the dialog box that appears, make sure Projection Type is set to Average ntensity and press OK k. Click and drag on the image to select a long cross-section rectangle passing through the central bright spot. The rectangle should be tall enough to include most of the central bright spot Analyze > Plot Profile Use the plot to determine the distance (in pixels) from the center of the central peak to each of the other maxima and minima. To convert the pixel distances (pixeldist) to angles, use the following conversion (note this conversion is good only when camera is at max optical zoom (digital zoom off) and camera focused at infinity (see also experiment 340): θ =!"#$%&"'(!.!"!"#!".! (4.)

11 Recall that equation 1 governs the interference pattern. Refer to table 1 which lists the values for m. θ =!!! Compare your experimental angle values calculated using the equation 4 with the theoretical values calculated using equation 1. Note that we are trying to use a commercial video camera as a poor man s photometer so that the relative intensity values recorded here are suspect. Professional scientific CCD cameras, advanced software and experimental analysis used in advanced lab courses may be needed to yield theoretically correct intensity values. Computer Simulations: Play with either the Adobe Flash simulation of an Airy disk (see web link on the lab wiki) or the CUPS simulations of circular diffraction on the lab computer. nstructions for the CUPS simulations are listed below. CUPS software can be used to create a colorful simulation of the Airy disk pattern To open CUPS, My Computer > Local Disk (C:) > Program Files > cups > CUPSWO > CUPSWO With the software now open, click nterference and Diffraction Then click Fraunhof > Circular Aperture Use the sliders to adjust the aperture radius and light wavelength o How does the pattern change if the wavelength is changed? Using the d plot, notice how wavelength affects the position of maxima and minima. Using the 3d plot, notice how wavelength affects the intensity of the maxima Do your observations agree with the theoretical equations governing the Airy disk pattern?

12 o How does the pattern change if the aperture radius is changed? How does the radius affect the position of the maxima and minima? How does the radius affect the intensity of the maxima and minima? Do your observations agree with the theory? How far does the pattern extend? At first, CUPS might only be displaying the first few orders of maxima and minima. Drag the rectangle around the coordinate grid located above the d plot. This will change which region of the interference pattern CUPS displays on the plots. o What happens as you look at regions further and further away from the central disk? Use the CUPS Fourier transform software and study the Fourier transform of a circular aperture and square apertures. Convince yourself that intensity of the Fraunhoffer diffraction pattern of a circular aperture is square modulus of the fourier transform of the circular aperture. Lens Diffraction Because a convex lens is thicker in its middle than near its edges, light passing through the middle of the lens will have to travel through more glass than light traveling through the edges. Therefore, if a beam of light is incident upon the entire lens, different parts of the beam will be diffracted by different angles depending on where they pass through the lens. This will create an observable diffraction pattern. The pattern is an Airy disk pattern resembling that created by a small circular aperture. 1. Set-up a. Place a laser at the left end of an optical bench and allow it to warm up for about an hour b. Attach a - mm lens to a component carrier and place it directly in front of the laser c. Attach a 48 mm lens to a component carrier and position it far enough away from the - mm lens that it is covered completely by the diverging beam ( mm is usually enough) d. Use the Gaussian lens formula* to calculate where the 48 mm lens will place the image created by the - mm lens i. The - mm lens places an image of the point source mm to its left (i.e. somewhere inside the laser) ii. The 48 mm lens then places an image of this image somewhere to its right (this is the distance to calculate) e. Use a fiber optic cable to connect a linear translator to a photometer f. Set the photometer sensitivity to.3 and connect the photometer to Science Workshop

13 g. Place the linear translator on the optics bench so that the tip of the fiber optic cable is the distance calculated in 1.d. away from the lens. Collecting data a. Open the Pasco Capstone software on the lab computer b. Click Create Experiment c. On the picture of the Science Workshop nterface, click on the input that the photometer is connected to and select Voltage Sensor d. Se the sample rate to 5 e. Close the Experiment Setup window f. n the Displays menu on the left side of the screen, double-click Graph g. Turn on the linear translator and then hit Start to begin gathering data h. Once enough data has been collected, press Stop and then turn off the linear translator By zooming in on the graph, you should be able to see the small peaks in addition to the large central peak of the Airy disk NOTE: The graph is of voltage vs. time so the greater the intensity of the light, the more negative the voltage. Thus, a positive peak in intensity corresponds to a negative drop in voltage. Here is a sample data plot. The central peak is too large to appear on the graph but the smaller first order maxima can be seen on either side of it

14 Fresnel Zone Plates A Fresnel zone plate uses diffraction to focus light to a single point. t consists of concentric rings of transparent and opaque material positioned at specific intervals to create maximum constructive interference at a point directly behind the center. Due to the constructive interference, the intensity of the focused light at this point is greater than the intensity of an unobstructed beam (i.e. one that does not pass through a zone plate). n this sense, the zone plate acts like a lens. 1. General set-up a. Position a red (63.8 nm) laser at the left end of an optics bench. t should warm up for at least an hour before the experiment can begin b. Create a beam expander i. Attach an 18 mm lens to a component carrier and position it as close to the laser as possible ii. Attach a 17 mm lens to a component carrier and position it exactly 144 mm ( = 144) to the right of the 18 mm lens iii. Attach a variable aperture to a component carrier and position it just to the right of the lenses c. Attach a zone plate to a component carrier and position it about 335 mm from the far right end of the optics bench d. Just past the far right end of the optics bench, place a linear translator on a lab jack and connect it to a photometer using a fiber optic cable e. Attach a 0.08 mm light source aperture to the front of the linear translator, as close to the tip of the fiber optic cable as possible. Precision set-up a. Make sure the expanded beam is incident on the entire zone plate i. The lenses will cause the beam to veer off to one side if the beam isn t passing directly through the centers of the lenses. Adjust the lenses to straighten out the beam ii. Adjust the variable aperture so that the beam has just about the same diameter as the zone plate. Having it be a little bigger than the zone plate is better than it being too small b. Position the light source aperture directly in front of the fiber optic tip i. Disconnect the fiber optic cable from the photometer and hold it under a lamp s light bulb ii. The end of the cable in the linear translator should now glow white. Adjust the slide with the aperture until you can see the cable glowing through from the other side

15 Brown University Physics Department PHYS 0160 LAB C The faint white dot indicates that the aperture is properly aligned in front of the glowing fiber optic cable c. Make sure the center bright spot of the beam is incident upon the light source aperture i. You should be able to observe the bright central spot of the diffraction pattern. f not, move the zone plate closer to or further away from the linear translator ii. Hold the photometer end of the cable under a lamp again to make the aperture glow white iii. Move the linear translator and use the lab jack to raise or lower it until the tiny center bright spot of the diffraction pattern is directly over the glowing-white aperture iv. Don t forget to reattach the cable to the photometer

16 Brown University Physics Department PHYS 0160 LAB C Although the image was saturated by the intensity of the laser, the glow of the fiber optic cable behind the aperture is visible as the small yellowish spot to the left of the red laser diffraction pattern. When properly aligned, both dots should share the same center Measure the intensity of the bright spot using the photometer. Compare this to the intensity of the unobstructed beam by adjusting the zone plate slide so that the beam passes through the transparent material under the opaque points on the other side of the slide. MPORTANT: be sure to compare the intensity of the zone plate bright spot to the intensity of the beam passing through the transparent material on the other side of the slide; DON T REMOVE THE SLDE ALTOGETHER. The transparent material is used in the zone plate and cuts down slightly on the intensity of light so you want its effect to be uniform throughout your data Proper set up for measuring unobstructed beam intensity 16