# TWO AND MULTIPLE SLIT INTERFERENCE

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1 TWO AND MULTIPLE SLIT INTERFERENCE Double Slit and Diffraction Grating. THEORY: L P L+nλ Light d θ L 0 C nλ Wall Screen P Figure 1 If plane waves of light fall at normal incidence on an opaque wall containing two narrow parallel slits a distance d apart, (Fig. 1), the light spreads out by diffraction upon passing through the slits. On a distant screen the overlapping beams from the two slits undergo interference, to produce a pattern of dark and bright fringes. At C, equidistant from the slits, all wavelengths of the light arrive in phase and interfere constructively to produce a central image or zero-order interference pattern having the same color as the original light. At some other point P which is at a distance L from one slit and L + nλ from the other (λ is some specific wavelength present in the light beam; n is an integer) there is also constructive interferences and a bright fringe appears with the color pertaining to that specific wavelength. At intermediate points distant L and L + (2n + 1) (λ/2), destructive interference occurs for that wavelength λ. If the original light beam contains only a number of well separated spectrum lines, each with its own λ (as happens in atomic spectra), the pattern on the screen is a set of lines for each λ, repeated below C and also repeated for numerous values of the integer n. The

2 pattern corresponding to n =1 is the first-order spectrum (one above and one below point C); n = 2 gives the second-order spectrum, and so on. It can be seen from Fig. I that the governing equation is n λ = d sin θ (1) provided d << L 0, which is ordinarily true. Therefore measuring the angle θ, counting n from the zero-order pattern, and knowing d in advance, we can calculate λ, the wavelength of the pertinent spectrum line. Doing this for several values of n increases the precision. Greatly increasing the number of slits (to hundreds or thousands), while keeping them all parallel and spaced at regular intervals d, produces a diffraction grating. Original gratings are made by ruling a glass surface with machines of great cost and complexity (each ruling makes an opaque strip on the glass; the slits are the undisturbed parts between rulings). Except in work of the highest precision, replica gratings are used. They are made by depositing a film of collodion, or the like, upon an original grating, stripping off the film, and mounting it on glass. The accuracy of the spacing d is not quite so good on a replica, but ample for a great many purposes. Substituting a grating for the two slits of Fig still keeping the total size of the grating much less than L 0 -- does not alter the form of Eq.1 for the bright fringes, or the meaning of the terms in it. The reason is that light traversing any pair of adjacent slits obeys Eq.1, and all such patterns are superimposed (provided that L o is very great). The availability of a great number of slits, N, means that the bright fringes are far brighter than two slits can produce. Also, each bright fringe becomes much narrower, for at an angle θ differing only very slightly from one that satisfies Eq.1 the light arrives from the various slits thoroughly out of phase. Therefore a grating far excels the two-slit arrangement, not only in brightness of the pattern but in resolving power. That is, the bright fringes are so narrow that two of them corresponding to spectrum lines of only slightly different wavelength can be told apart or resolved. This is important, for many atomic spectra are extremely complicated. In this experiment the diffraction from various simple arid multiple slits is investigated. To make quantitative measurements of the diffraction patterns a spectrometer is used (see Fig.2). Light from a mercury source is concentrated on the collimator slit by means of the condensing lens C. The collimator produces a beam of parallel light, which is incident on one of the slits or gratings on the photographic slide. The diffraction pattern is observed using the telescope; the angular position of which may be found using the vernier scale. The Pasco 9165-A photographic slide has a series of single slits of widths 0.02, 0.04, 0.08, and 0.l6 mms. The Pasco 9165-B slide is a series of double slits consisting of 0.04 and 0.08 mm slits with their centers and mms apart. For the Pasco 9165-C

3 slide there are gratings consisting of 2, 3, 4, and 5 slits each 0.04 mm wide and with their centers mms apart. The mercury source produces three principal spectral lines and by the use of filters any one of these may be separated out from the rest of the spectrum sufficiently well for the purposes of this experiment. Since the mercury source is fairly powerful and produces a certain amount of ultraviolet, you should avoid looking at it directly. PROCEDURE: 1. By viewing a distant object adjust the te1cscope so that it is focused to receive parallel light. The eyepiece must also be adjusted so that the cross wires are in sharp focus. 2. Place the red filter behind the slit of the collimator and using the telescope, view the light coming from the collimator. Do not insert the photographic plate into the beam at this stage. Adjust the condenser lens, collimator slit and collimator focus control until a sharply focused narrow upright slit is visible in the telescope. 3. Investigate the diffraction produced by the sequence of single slits. Notice how the pattern changes when a different filter is used. Determine the angles at which a given minimum or maximum occurs in the diffraction pattern for different filters. Notice the effect of slit width. Filter Color and Given Wave Length d n Measured θ Calculated θ For each of the 3 filters, measure the positions of several maxima or minima for one slit width. Using the given wavelengths and Eq. 1, calculate the expected positions of these maxima or minima, and compare these with your measured position. 4. Investigate the diffraction produced by the double slits. Describe qualitatively the patterns produced and how they vary with the filter used and the separation of the slits. For each of the filters and for 1 slit separation, carry out the measurements and comparisons described in Step Observe the patterns produced by the series of gratings. Notice how the pattern changes when the number of lines increases and when different filters are used. For each filter and one grating, carry out the measurements and comparisons described in Step Carry out the double slit experiment using the laser (green and red lasers are

4 Question: available) and LEDs (480, 560, 590, 635, and 665 nm LEDs are available). Do not look into the Laser Beam. See Method 2, Computer Acquisition with Digital Video below for sample data. (i) How do you think the results of this experiment would be changed if an incandescent filament lamp were used instead of the mercury source? Mercury Source Condenser Lens Filter Collimator Photographic Plate Telescope Fig. 2 Note: The grating and spectrometer method will be used in a later experiment determine the wavelengths of some of the lines of hydrogen spectrum

5 METHOD 2 Double Slit: Computer Acquisition with Digital Video You can also investigate double slit diffraction using a video camera and imaging software. By analyzing the intensity of the pattern with video software VirtualDub and ImageJ the wavelength of the light source can be measured. You can investigate both double and single slit diffraction patterns with this method and calculate the wavelength of your light source. For single slit diffraction, minima location is governed by the following Equation (1). Theta is the angle from the 0 th order maxima, n is the order of the minima, lambda is the wavelength, and b is the width of the slit. b sin θ = nλ (1) The governing equation for maxima location is equation (2). 1 b sinθ = n + λ (2) 2 For double slit diffraction, the governing equation for maxima location is (3), where d is the distance between the slits. d sin θ = nλ (3) To help improve your intuition about diffraction, you can further explore the phenomenon with the CUPS diffraction software. From the desktop, open the cups folder and double click the waves and optics icon. Select interference and diffraction from the main menu. Here you can adjust various parameters and see their effect on the intensity profile. For more information about the interference and diffraction simulation software and suggested simulations see the chapter Interference and Diffraction by Robert A. Giles pp PROCEDURE: 1. Align the spectrometer with the eyepiece, bringing the diffraction pattern into full view. Make sure there is no doubt as to which pair of slits the light is passing through. Record the slit spacing. 2. Swing the eyepiece of the spectrometer to one side so it isn t obstructing the path of the light. 3. Be sure the camera is focused at infinity. If necessary, focus it on an object as far away as possible, possibly down the hallway. Disable the digital zoom

6 4. Position the camera directly in front of the diffraction grating and adjust its position until the diffraction pattern is visible in the eyepiece. Use the maximum possible optical zoom and center the pattern. 5. Record a short video with Virtual Dub and open it in Image J. (the Video Capture Instructions below explain the capture and analysis in greater detail) 6. Use the averaging algorithm to smooth the image. 7. Draw a thin box around the center of the diffraction pattern. For maximum accuracy, do not extend the box beyond the visible pattern. 8. Select plot profile under analyze, and record the positions of the maxima in pixels. 9. To determine the conversion factor, pick one of the maxima. Swing the edge of the camera a couple degrees past the maximum and then return (to minimize backlash) so that it is at the edge of the field of view. Record the angular position. Continuing in the same direction, sweep back across the field of view until the maximum is aligned with the opposite edge of the screen. Record the angular position and calculate the number of degrees in one view. Divide this number by 720 pixels and convert to radians to find the number of radians per pixel. (Note that the rotation stage used in this part may not be available due to its use in other advanced experiments, if it is not available the following conversion will be good if you set the Canon Optura digital camera to max optical zoom (digital zoom off) 3.21mm θ ( radians) = ( pixeldiff / 720). Where 3.21 mm is the CCD width and 67.2mm 67.2 mm is focal length of the camera at max optical zoom). 10. Calculate the distance between each peak and the zeroth peak in pixels, and convert to radians. Calculate the wavelength. 11. Use various filters to measure a few different wavelengths. You can also use different light sources. With the same apparatus, replace the mercury lamp with an LED board. There are multiple colored LEDs per board so you can measure multiple wavelengths. You may not be able to image the pattern using the same magnification you used for the mercury lamp. If you change the zoom at any point during the alignment, just recalibrate the camera (step nine). 12. A third light source option is to use a laser and an optical bench. The light should pass from a laser through a polarizer to reduce the intensity, through the diffraction grating, to the camera. Again, you will probably have to adjust the zoom and radian/pixel calibration. Note: This apparatus source is particularly prone to focusing error so make sure the camera is focused at infinity. Video Capture Instructions: A. To capture video open VirtualDub.exe i. Click: file > capture avi

7 ii. Click: file > set capture file (remember where you put it) iii. Click: audio > uncheck enable audio capture (Don t forget to do this!) iv. Click: capture > capture video v. After about ~one second, hit esc to end the capture. You should have captured 30 to 60 frames (closer to 30 preferred) and the audio size should be 0 bytes. vi. Click: file > exit capture mode vii. Click: file > open video file (open your file) viii. Click: file > save as avi ix. Change the name of the file and save it x. Exit VirtualDub B. To obtain data from the image open WCIF ImageJ i. Click: file > open (open the one with the new name) ii. Click: plugins> stacks - z-functions > grouped ZProjector iii. Leave group size alone, but change projection type to average intensity. iv. The program will process and eventually open the averaged image. The image quality should be much sharper which will make measurement of the ring radius easier. v. With the mouse draw a thin horizontal box across the center of the image. The height should be small enough to minimize the effect of the curvature and the length should be long enough to include a number of rings on each side of the center. vi. Click: analyze > plot profile

8 Sample Data: Fig. 1 Image of a diffraction pattern from a mercury lamp (yellow filter 578 nm) through slits of width 0.08mm and spacing 0.250mm. Fig. 2 ImageJ plot profile of a diffraction pattern from a mercury lamp (yellow filter 578nm) through double slits of width 0.08mm and spacing 0.250mm. Sample Student Data : Double Slit, Yellow Filtered Light (578nm) Separation Maxima Order Location(pixels) from 0 th Order Theta (rad) Wavelength(nm) Mean: Degrees in the field of view: Degrees per pixel: Radians per pixel: D=250,000nm

9 Fig. 3 Image of a diffraction pattern from a 560 nm LED through a double slit of width 0.08mm and spacing 0.250mm. Fig. 4 Image J plot profile of a diffraction pattern from a 560 nm LED through a double slit of width 0.08mm and spacing 0.250mm. Sample Student Data :LED Double slit, 560nm Separation Maxima Order Location(pixels) from 0 th Order Theta (rad) Wavelength(nm) Mean:

10 Fig. 5 Image of a diffraction pattern from a red laser through a double slit of width 0.08mm and spacing 0.250mm. Fig. 6 Image J plot profile of a diffraction pattern from a red laser (632.8nm) through a double slit of width 0.08mm and spacing 0.250mm. Sample Student Data: Double slit, Laser 630nm Separation Maxima Order Location(pixels) from 0 th Order Theta (rad) Wavelength(nm) Mean:

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