Physics lab Hw
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interferf12.pdf
Phys 212/216L Experiment Interference and Diffraction
Spring 2011 1
Figure 1 Geometry of Two-Slit Interference
I. Introduction and Objective This lab1 focuses on the nature and behavior of light. The wave properties of light are most easily demonstrated by interference and diffraction of light as it passes through narrow slits. The wave nature of light results in a pattern with a series of bright and dark regions related to the wavelength of the light and the number and size of the slits. You will examine the interference and diffraction patterns created by performing single and double slit experiments using two different lasers. By analyzing the double slit interference patterns, you will determine the wavelength of each source. Then using the known wavelengths of each laser, you will analyze single slit diffraction patterns and determine the slit width. You will also compare the interference pattern produced in the two slit experiment to the diffraction pattern produced in the single slit experiment. II. Theory Huygen’s principle can be used to explain the patterns formed when light passes through either a double or single slit. For two-slit interference, each slit acts as a new source of light. Since the slits are illuminated by the same wave front, these sources are in phase. Where the wave fronts from the two sources overlap, an interference pattern is formed. If you look closely at a two slit interference pattern, you will notice that the intensity of the fringes varies. a) Two Slit Interference In 1801, Thomas Young demonstrated the wave nature of light by observing the pattern formed when a narrow beam of light was sent through a pair of narrow slits. In a similar way, if a laser beam is passed through two vertical narrow slits onto a distant screen, a horizontal pattern of equally spaced dots is observed. The bright and dark regions are due to the constructive and destructive interference of the light waves from the two slits. The geometry of the double slit setup is shown in the Figure 1 where d is the slit separation, L is the distance from the slits to the screen, and y is the distance measured from the center of the pattern. When the distance which the light beams travel from the two slits to the screen differs by one-half wavelength, the two beams will cancel, and a dark region (a minimum) will be observed. When the screen is far from the slits (the length L is much greater than the slit separation, d), the positions of these minima can be given by
d L
ny λ
)( 2 1+= ,...2,1,0 ±±=n (Dark fringes of a double slit) (1a)
and the position of the interference maxima are given by
d L
ny λ
= ,...2,1,0 ±±=n (Bright fringes of a double slit) (1b)
where n is an integer, and λ is the wavelength of the light. It can be shown from equation (1a) that the minima in the light are equally spaced with a spacing W given by
d L
W λ
= (2)
1 Turner, Clemson University General Physics Laboratory Manual, 1995.
Phys 212/216L Interference and Diffraction Name _______________________
Fall 2012 2
This distance W referred to as the width of the interference peak. It is a constant for all peaks in this interference pattern, as indicated in Figure 1. b) Single Slit Diffraction When a beam of light passes through a narrow vertical single slit, it is observed to spread out horizontally. This bending of the beam as it passes through a narrow opening is known as diffraction. The diffraction pattern is also observed to have a series of maxima and minima due to the interference effects of the light passing through different portions of the single slit. The geometry of this single slit setup is shown in the Figure 2, where b is the width of the slit. The central maximum is by far the brightest part of the pattern. When the distance to the screen, L, is much greater than the slit width, b, the positions of the minima in the single slit diffraction pattern can be given by
b L
ny λ
= ...3,2,1 ±±±=n (Dark fringes for single-slit diffraction) (3a)
And the position of the diffraction maxima are given by
b L
ny λ
)( 2 1+= ...3,2,1 ±±±=n (Bright fringes for single slit diffraction) (3b)
The distance Wo between the minima on either side of the central peak is found from equation (3a) to be twice the distance W between the other adjacent minima.
b L
Wo λ2
= Central Peak (4a)
b L
W λ
= All other peaks (4b)
The effect of the single slit diffraction is also observed in the double slit interference pattern. When the light beam passes through the slits, it spreads horizontally due to diffraction. The reason that the double slit pattern gradually decreases in intensity is due to the decrease in intensity of the light coming from the two slits.
III. Experiment Apparatus The equipment needed includes: a diffraction plate (a diagram of the diffraction plate is shown in Figure 3), a component holder, a red and green HeNe laser, viewing screen/chart, a ruler, and meter stick, a stand with clamp to hold the green HeNe laser and optics bench (optional) to place the red HeNe laser.
Figure 2 Geometry of Single-Slit Diffraction
Phys 212/216L Interference and Diffraction Name _______________________
Fall 2012 3
CAUTION! Always use extreme care when the laser is turned on! Never look directly at it the beam, or point it at another person. a) Two-Slit Interference
Procedure The purpose of this part of the experiment is to examine the light patterns obtained from double slit interference, and to use this to determine the wavelength of the light emitted from each of the lasers. The equipment is set up similar to the schematic shown in Figure 4a. The laser is placed on the table so that it the light beam will hit the paper screen placed at the other end of the table. Tape a sheet of paper over the screen so that you can record your measurements of the pattern produced on this sheet.
Attach the diffraction plate to one of the component holder, as shown in Figure 4b, and place it in front of the laser with the slide perpendicular to the beam. Center one of the double slit patterns such as D (E or F) in the laser
Figure 4(a) Schematic of Equipment Setup
Diffraction Plate
Figure 4(b) Diffraction plate on a component holder; placed on an optics bench
Figure 3 Diffraction Plate Apertures
Phys 212/216L Interference and Diffraction Name _______________________
Fall 2012 4
beam. When adjusting the position of the slit in the beam, view the slide from behind as indicated in the setup figure. You should observe a series of bright spots on the screen. You may need to move the slide holder back and forth slightly to get the best pattern. Once it is adjusted satisfactorily, be careful not to move anything while making your measurements. Your interference measurements will be made directly on the paper on the screen. Mark the positions of the minima of the light intensity pattern. That is, mark the position of as many dark spots as possible with a short vertical line, estimating the location of center of each of the minima. Be careful not to move the laser while doing the following. Measure the distance L from the diffraction plate to the screen. Remove the plate and holder, and note the location of the laser beam on the chart. Mark it with an arrow. Use the section labeled data and results to record your results. Remove the chart from the screen to make measurements of the interference pattern. Count the number of spots measured. Divide the total length of the pattern measured by the total number of spots to get the average width, W, of each peak. Verify that the width of the central maxima Wo is about the same as all other peaks. From the information listed in Figure 3, record the slit used and the slit spacing, below. Data and Results Part a1) Two-Slit Interference: Wavelength calculation using first laser–
Laser used, circle one: red; green Slit used ______________ Slit separation, d, = ______________ Distance L, from the plate to the screen = ___________ Total length of pattern measured, l’ = _____________ Total number of spots measured = ______________ Average Width W for one peak = ______________ Width W0 for the central peak = ______________ Experimental wavelength, λ = _____________ Actual wavelength, λred = 633 nm, λgreen = 532 nm Percent error = _________________ Part a1) Two-Slit Interference: Wavelength calculation using second laser–
Laser used, circle one: red; green
Figure 5 Double slit interference pattern
l'
Phys 212/216L Interference and Diffraction Name _______________________
Fall 2012 5
Slit used ______________ Slit separation, d, = ______________ Distance L, from the slit to the screen = ___________ Total length of pattern measured, l’ = _____________ Total number of spots measured = ______________ Average Width W for one peak = ______________ Width W0 for the central peak = ______________ Experimental wavelength, λ = _____________ Actual wavelength, λred = 633 nm, λgreen = 532 nm Percent error = _________________ ------------------------------------------------------------------------------------------------------------------------------------------ Use the measurements and the information you recorded above, the average spacing W, and equation 2 to calculate the wavelength of the laser. 1a) Show your work and calculate λ for the first laser used. b) The known wavelengths are 633 and 532 nm for the red and green HeNe lasers, respectively. One way to determine how accurate your experimental measurements are is to calculate a percent error. To calculate percent
error, one uses the following equation: %100% × −
= accepted
measuredaccepted error
λ
λλ . Calculate the percent error in
your measurement of the wavelength of this laser. 2a) Show your work and calculate λ for the second laser used. b) Calculate the percent error in your measurement of the wavelength of this laser.
Phys 212/216L Interference and Diffraction Name _______________________
Fall 2012 6
3) How does the spacing for the central peak compare to the average spacing for the other peaks? Is your answer and therefore measurements consistent with equation (2)? b) Single Slit Diffraction The purpose of this part of the experiment is to examine the light pattern obtained from single slit diffraction, and to use this to determine the width of the slit. For the remainder of the experiment, consider the wavelength of the lasers to be the known values of 633 nm and 532 nm for the red and green laser, respectively. Procedure The setup is the same as that used above for the double slit interference experiments (Figures 4). Again, tape a sheet of paper to the screen so that the laser beam is in the center of the paper. Adjust the slide so that the laser is shining through one of the single slit patterns such as slit C (B or A). Mark the positions of the minima as before. Also, remeasure the distance L from the diffraction plate to the screen. Remove the paper from the screen to analyze the results. Measure the central peak width Wo, and measure the width of at least two other peaks. Record the slit, its width, and the laser used in the data and results section, below. Data and Results Part b1) Single Slit Diffraction – slit width calculation using first laser Laser used: ____________________ , λ = ____________________ Slit used ______________ Nominal slit width, b = _________________ Distance L, from the slit to the screen = ___________ Width W0 for the central peak = ______________ Total Width of side peaks = ______________ Number of side peaks = ______________ Average width W of side peaks = ______________ Experimental slit width, b = _________________ Percent error = _______________
Figure 6 Single slit diffraction pattern
Phys 212/216L Interference and Diffraction Name _______________________
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Part b2) Single Slit Diffraction – slit width calculation using second laser Laser used: ____________________ , λ = ____________________ Slit used ______________ Nominal slit width, b = _________________ Distance L, from the slit to the screen = ___________ Width W0 for the central peak = ______________ Total Width of side peaks = ______________ Number of side peaks = ______________ Average width W of side peaks = ______________ Experimental slit width, b = _________________ Percent error = _______________ ------------------------------------------------------------------------------------------------------------------------------------------- Use the measurements and information recorded above, the known wavelength of the laser and equation (4b) to calculate the slit width, b; refer to Figure 2. 4a) Show your work and calculate the slit width. b) Then check the chart in Figure 3 to determine the nominal width for the slit you used.
• Slit width =
• Check the accuracy of your measured value of the slit width, by calculating a percent error.
5a) Show your work and calculate the slit width. (Second laser used) b) Then check the chart in Figure 3 to determine the nominal width for the slit you used.
Phys 212/216L Interference and Diffraction Name _______________________
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• Slit width =
• Check the accuracy of your measured value of the slit width, by calculating a percent error.
Repeat the above experiments using the other laser. That is go back to part a) the double slit experiment and make the necessary measurements to calculate the wavelength of this laser. Then repeat part b) the single slit experiment, with the second laser, making the necessary measurements to calculate the slit width used. Record all pertinent information in the data and results section. 6a) Compare the similarities and differences of the two diffraction patterns produced and determine if they verify the relationship given in equations (3) and (4). b) How does the width of the central peak compare to the other peaks? c) How does the pattern change as one goes to a wider/narrower slit (consider the spacing between dark fringes)? Are your all answers and therefore your measurements consistent with equations 3 and 4? V. Conclusions In today’s lab, you studied of the wave nature of light. You performed two experiments, Two Slit Interference and Single Slit Diffraction. The resulting light/intensity patterns formed are a direct consequence of the wave behavior of light and are related to the wavelength of the light and the number and size of the slits. By examining the interference patterns produced in the Two Slit experiment, you were able to determine the wavelength of the lasers used. Then using the known laser wavelengths and by examining the diffraction patterns produced in the Single Slit experiment you determined the width of slits used. A comparison was made between the intensity patterns produced in the case of interference and that of diffraction, noting the width of the central maxima and other peaks, thereby verifying equations (1) – (4). Homework 1) Consider the spacing and peak widths of the light patterns produced by diffraction and interference of light. a) The spacing of the minima in the patterns produced by a green laser will be: greater than, less than or equal to that of a red laser. Justify your answer. b) The width of the peaks, W and/or Wo, of the pattern for the green laser will be: greater than, less than or equal to that of a red laser. Justify your answer.
Phys 212/216L Interference and Diffraction Name _______________________
Fall 2012 9
2) Light from a HeNe laser (λ = 632.8 nm) strikes a pair of slits at normal incidence, forming a double slit pattern on a screen 1.4 m from the slits, as shown in the figure below. What is the slit spacing? 3) An experiment is performed by passing light through one or two slits and the light pattern to the right is produced. Determine the best answer and explain how you can tell:
a) the pattern produced was a result of a double slit
interference experiment
b) the pattern produced was a result of a single slit diffraction experiment
c) More information is needed, i.e. the wavelength of the source, the distance it is from the slit(s) and how far the
slit is to the viewing screen, before it can be determined which type of experiment produced the pattern
4) Now, you observe the following light pattern. Select all the possible changes in experimental conditions that might have caused the differences in the original pattern (that shown in problem 3) to that which you observe now. a) the wavelength of the light source was increased
b) the wavelength of the light source was decreased
c) the slit width/slit separation was increased
d) the slit width/slit separation was decreased
e) the experiment was changed from a double slit to single slit
f) the experiment was changed from a single slit to a double slit
23.0 mm
