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+ | <html><STYLE> #jsMath_Warning {display: none} </STYLE></html> | ||

+ | ===== PHY 124 Lab 6- Interference and Diffraction ===== | ||

+ | |||

+ | [[http://www.ic.sunysb.edu/Class/phy122ps/labs/dokuwiki/pdfs/124lab6worksheet.pdf|Important! You need to print out the 3 page worksheet you find by clicking on this link and bring it with you to your lab session.]] | ||

+ | |||

+ | If you need the .pdf version of these instructions you can get them [[http://www.ic.sunysb.edu/Class/phy122ps/labs/dokuwiki/pdfs/phy124lab6.pdf|here]]. | ||

+ | ===== Goals ===== | ||

+ | |||

+ | The purpose of this laboratory is to study the phenomena of diffraction and interference. In Part I, you will observe the diffraction of light by a human hair (similar to the diffraction on a narrow slit), and in Part II, both interference and diffraction of light by various types of slit arrangements. You can do either Part I or Part II first. | ||

+ | |||

+ | ===== Video ===== | ||

+ | <flashplayer width=640 height=480>file=http://www.ic.sunysb.edu/Class/phy122ps/labs/phy122vid/phy124lab6.flv</flashplayer> | ||

+ | |||

+ | ===== Part I - Diffraction of monochromatic coherent laser light by a human hair. ===== | ||

+ | |||

+ | In Part I, you will observe the diffraction of light by a human hair (similar to the diffraction on a narrow slit), calculate the hair diameter from your observed diffraction pattern and compare it to a measurement done with a micrometer. | ||

+ | ==== Equipment for Part I ==== | ||

+ | |||

+ | {{124l6fig1.jpg?600}} | ||

+ | |||

+ | * Helium-Neon Laser with a wavelength of 632.8 nm. | ||

+ | * Slide Frame to mount the hair. | ||

+ | * Paper to display the diffraction pattern on the wall. | ||

+ | * Masking tape to record the pattern. | ||

+ | * Ruler | ||

+ | * Micrometer | ||

+ | ==== Reading the micrometer ==== | ||

+ | |||

+ | {{124l6fig2.jpg}} | ||

+ | |||

+ | A micrometer can be used to measure very small distances and diameters. Each division on the knob of the micrometer represents 10 μm, or 0.01 mm. Each division along the handle represents 0.50 mm. When measuring an object, turn the knob until the micrometer is fitted around the object. Then read the number of lines visible on the long part of the handle. This is how many half mm you have. Then, find which marking on the knob lines up with the center of the line on the handle. Multiply this number by 0.01 to get the amount in mm. Then add it to the number of half mm to get your reading. You can consider the measurement error on this device to be equal to one of the divisions on the knob (i.e 10 μm) | ||

+ | |||

+ | Example: You measure the diameter on a pencil. After fitting the micrometer around the pencil you see that there are 12 lines visible on the long part of the handle, which gives a value of 6mm. Then on the knob you find that the 45 mark lines up with the line going down the handle. Multiplying 45 by 0.01 gives 0.45 mm. Adding this to the first number gives you a diameter of 6.45±0.01 mm for the pencil. | ||

+ | ==== Measurement of the hair diameter with a micro-meter ==== | ||

+ | |||

+ | First, you will determine the zero diameter reading so you have a clear reference point when measuring the diameter of the hair. Close the micro-meter down without the hair and close the micrometer by applying about the same force you will use later when you are measuring the hair. (The micrometer will show different “zero” readings whether you close it gently or with a large force). What value do you read off of the micrometer? This will be your zero value. When measuring the diameter of the hair remember to start counting from your zero value, which might not actually read 0 mm on the micrometer. Record this zero reading on your worksheet. The scale of the micro-meter is shown in Fig 2 above. | ||

+ | |||

+ | Insert the hair, which you can supply from your own head, close down on it with the micro-meter screw and record your reading on your worksheet. | ||

+ | |||

+ | Subtract the zero reading from the reading with the hair to obtain the hair diameter. Include the unit and your estimate of the error of the hair diameter. | ||

+ | ==== Measurement of the hair diameter from the diffraction pattern ==== | ||

+ | |||

+ | {{124l6fig3.jpg}} | ||

+ | |||

+ | Tape the hair onto the slide frame in a vertical position. Put the laser ~ 1.2 m from the wall and align its axis perpendicular to the wall. Place the slide frame ~ 10 cm from the laser. These are not the exact values you will use for this lab. You will need to make slight adjustments to your individual set up. When recording your distances in your worksheet remember to accurately measure the distances for your set up. | ||

+ | |||

+ | Move the slide sideways until the laser beam hits the hair fully. When this occurs you should see the hair light up from the laser hitting in and a diffraction pattern should appears on the wall. The central, very bright, roughly circular spot is due to the laser beam being much wider than the hair. This light is not being diffracted and is hitting the wall directly. Ignore this spot and concentrate on the wider and less intense central diffraction maximum which is spread out horizontally. | ||

+ | |||

+ | === Qualitative observations === | ||

+ | |||

+ | You will see that diffraction on a hair is very similar to diffraction by a narrow slit (see Ch 21 sheet 25). (The equivalence of the two diffraction patterns is the subject of Babinet’s Theorem – not discussed here.) | ||

+ | |||

+ | - Record on your worksheet the direction of the spread of the diffraction pattern when the hair is positioned vertically. Turn the frame by 90 degrees so that the hair is horizontal. Move the frame vertically until the laser beam is obstructed by the hair and the diffraction pattern in observed. Record the direction of the pattern seen while the hair is in positioned horizontally. | ||

+ | - Put the hair back in the original, vertical position. Shift the frame by ~ 0.5 m closer to the wall. Observe and record what happens to the pattern, specifically the positions on the intensity maxima, on your worksheet. How are D and x and related? (Use your observations from the cases where the hair is ~ 10 cm from the laser and from where the hair is ~ 0.5 m from the laser.) | ||

+ | |||

+ | |||

+ | === Quantitative measurements === | ||

+ | |||

+ | In this part of the lab you will use the diffraction pattern and the geometry of your experimental set up to obtain a value for the diameter of the hair. | ||

+ | |||

+ | Record the wavelength of the laser light on your worksheet. | ||

+ | |||

+ | Again, position the frame with the hair close to ( ~ 10 cm) the laser. Measure the exact distance D between the frame and wall and estimate its error ΔD. Record these values on Execution Sheet 2. Place a long piece of masking tape (from your TA) so that the diffraction pattern appears on the tape. Mark the center positions of the diffraction minima (darkness) to the left and the right of the central maximum. Place the tape on your worksheet. | ||

+ | |||

+ | For the first 5 orders of diffraction (m = 1,2,3,4,5) measure the distance (2x) between the corresponding minima on either side of the beam. Measure from center to center of the diffraction minima and enter these values into the table on your worksheet. Estimate for m ~3 the error of (2x). This will be the error you use for each value of 2x. | ||

+ | |||

+ | BE CAREFUL! When you make your experimental measurements you are measuring the distance from the center of one minimum on one side of the central maximum to the center of the same order minimum on the other side of the central maximum. This value is 2x ± Δ(2x). When doing your calculation you will only use the distance from the center of a minimum to the center of the central maximum, or x ± Δx. | ||

+ | |||

+ | You now need to use the tool below to plot your data and find the diameter of the hair. | ||

+ | |||

+ | From values in the table on your worksheet find the distance x for each m value and enter it in the table below. Include values for the error in x. When you click submit you get a plot of your distances x versus the order m of the diffraction minimum for the 5 measured values of x. Record the slope of the graph and it's error on your worksheet, we will use this to find the hair diameter. | ||

+ | |||

+ | <html> | ||

+ | <form method="post" action="http://mini.physics.sunysb.edu/~mdawber/plot/basicplot2.php" target="_blank"> | ||

+ | x axis label (include units): <input type="text" name="xaxis" size="60"/><br> | ||

+ | y axis label (include units): <input type="text" name="yaxis" size="60"/><br> | ||

+ | |||

+ | <input type="hidden" name="zero" value="y" /><input type="hidden" name="errortype" value="y"> | ||

+ | |||

+ | m= <input type="text" name="x1" value="1" size="10"/>    x= <input type="text" name="y1" size="10"/>+/-<input type="text" name="dy1" size="10"/><br/> | ||

+ | m= <input type="text" name="x2" value="2 "size="10"/>    x= <input type="text" name="y2" size="10"/>+/-<input type="text" name="dy2" size="10"/><br/> | ||

+ | m= <input type="text" name="x3" value="3" size="10"/>    x= <input type="text" name="y3" size="10"/>+/-<input type="text" name="dy3" size="10"/><br/> | ||

+ | m= <input type="text" name="x4" value="4" size="10"/>    x= <input type="text" name="y4" size="10"/>+/-<input type="text" name="dy4" size="10"/><br/> | ||

+ | m= <input type="text" name="x5" value="5" size="10"/>    x= <input type="text" name="y5" size="10"/>+/-<input type="text" name="dy5" size="10"/><br/> | ||

+ | <input type="submit" value="submit" name="submit" /> | ||

+ | </form> | ||

+ | </html> | ||

+ | |||

+ | The equation for diffraction minima is given in Ch21 on sheet 25, using this equation the small angle approximation, and the geometry in Fig. 3, the diffraction minima can be expressed in the following way: | ||

+ | |||

+ | <WRAP column 35%>\\ | ||

+ | </WRAP> | ||

+ | <WRAP column 45%> | ||

+ | $\Large m\lambda=b\sin \theta \approx b\tan \theta \approx \frac{bx}{D},m=1,2,..$ | ||

+ | </WRAP> | ||

+ | <WRAP column 10%> | ||

+ | (6.1) | ||

+ | </WRAP> | ||

+ | \\ | ||

+ | |||

+ | where b is the hair diameter, λ the wavelength, D the distance hair-wall ,θ the angle between forward direction and a line pointing to the diffraction minimum of order m, and x the distance on the screen between θ-0and the m-th diffraction minimum. Use the equation given above to write x = slope * m and express b in terms of the measured slope, λ and D. Propagate your errors of D and the slope into the error of b. (Use expression E.7 in Error and Uncertainty.) Record the value you obtain for the hair diameter on your worksheet and compare it to the one you measured with the micrometer. Are the two values consistent within error? | ||

+ | |||

+ | =====Part II ===== | ||

+ | |||

+ | In Part II you will observe both interference and diffraction of light by various types of slit arrangements and explain the patterns observed. | ||

+ | ==== Equipment for Part II ==== | ||

+ | |||

+ | |||

+ | {{124l6fig4.jpg?600}} | ||

+ | |||

+ | | ||

+ | Incandescent Light Bulb with vertical filament sitting in a box covered with red (bottom) and blue (top) filters. | ||

+ | |||

+ | 3” x 5” Glass Slide with various slit arrangements. | ||

+ | |||

+ | ==== Diffraction and Interference of red and blue light through various slit arrangements ==== | ||

+ | |||

+ | Orient the slide such that you see the double slit in the bottom right corner of the slide as shown in Fig 4. Identify the 5 columns of slit arrangements with column A,B,C,D,E, and the rows with 1,2,3,4,5 (like in a spreadsheet) as shown in Fig 4. | ||

+ | |||

+ | You will not use columns B and D. There is only one light box for this part of the lab which you will be sharing with the rest of your section, however there will be several sets of slits so many students can view the box at once. You should record your observations in a timely manner so that everyone can successfully complete this part of the lab. | ||

+ | |||

+ | You will be observing the red and blue filaments of the light bulb. Hold the slide close to your eyes and observe the red and blue-filtered filament through the various apertures as follows. | ||

+ | |||

+ | === Single Slit Diffraction === | ||

+ | |||

+ | Column A has 5 rows of single slits with the slit width b decreasing by a factor of 2 as you step through the rows from top to bottom. | ||

+ | |||

+ | Start at A5. You should see a single slit diffraction pattern with a broad central intensity maximum and side maxima which become weaker as you go away from the center. As you saw in Part 1, the intensity maxima alternate with intensity minima. | ||

+ | |||

+ | What you are looking at is the intensity distribution as sketched in Ch21 sheet 25, (the dashed blue line. | ||

+ | |||

+ | Starting from A5 go up the rows on the slide and observe how the central diffraction maximum and the distances of the minima from the center vary. Refer back to your Lab Prep exercise where you looked at the behavior of the formula $b\sin \theta = m \lambda$ . Does what you see match with what the formula predicts? | ||

+ | |||

+ | Pay attention to the following points: | ||

+ | |||

+ | * Is the blue pattern, wider or narrower than the red one? Why? | ||

+ | * How do the central maximum and the minima vary with slit width b? | ||

+ | * Why does the pattern eventually disappear when b becomes very large (b >> λ)? | ||

+ | |||

+ | (Hint: what happens to θ when b >> λ, do the minima spread apart or crowd together?) | ||

+ | |||

+ | === Double Slit Interference === | ||

+ | |||

+ | NOTE! For what follows, recall that when studying two or multi slit interference, single slit diffraction is always present due to the finite size of the slit width, and that the single slit diffraction intensity distribution (red dashed curve on sheet 26 in Ch21) controls the intensity distribution of the interference pattern as is illustrated on sheet 26 in Ch21. | ||

+ | |||

+ | E1 has the same single slit as A5 and thus produces the same pattern. Positions E2 to E5 have 2 slits with the slit width b kept constant and the same as E1 and the slit distance d increasing by a factor of 2 for each step down through the rows. | ||

+ | |||

+ | Observe the light from farther back. As you go from E1 to E2, you should notice how the central broad diffraction maximum (one bright band in E1) is now divided into 3 narrower bright lines in E2. This is the same for red and blue light. Starting from E2 go down the rows on the slide and observe how the number of maxima (or minima) inside the central diffraction maximum changes. Note also when the two slit interference maxima/minima become difficult to observe. | ||

+ | |||

+ | Recall your Lab Prep exercise which looked at the behavior of the equation $m\lambda=d\sin\theta$. Does what you see match with what the formula predicts? | ||

+ | |||

+ | As you go from E2 to E5 how does the number of two slit interference maxima (or minima) inside the central diffraction maximum change? | ||

+ | |||

+ | The slit widths are the same for all slits in column E. What does this mean for the width of the central diffraction maximum? | ||

+ | |||

+ | === Many Slits, going toward the Diffraction Grating === | ||

+ | |||

+ | Column C has from 15 to 80 slits in the various rows. The slits are distributed over the same size area for each row on the slide, so the separation between slits changes depending on the number of slits in each row. | ||

+ | |||

+ | The slit width varies as well, so ignore the width of the central diffraction maximum and concentrate on how much the many-slit interference maxima are spread apart as you go from row to row. | ||

+ | |||

+ | You should notice that the spread changes as you go from C1 to C5 but does not change uniformly. | ||

+ | |||

+ | Explain your observations about the change in the spread of the maxima from slide to slide in terms of the equation $m\lambda=d\sin\theta$ using the fact that different numbers of slits are etched into the same size area on the different slides. |