Interference and Diffraction of Light

Transcription

1 [International Campus Lab] Objective Observe interference and diffraction patterns for various slits and diffraction gratings, and find the wavelengths of laser sources. Theory Reference Young & Freedman, University Physics (14 th ed.), Pearson, Interference and Coherent Sources (p.1184~1187) 35.2 Two-Source Interference of Light (p.1188~1191) 36.2 Diffraction from a Single Slit (p.1212~1215) 36.3 Intensity in the Single-Slit Pattern (p.1215~1219) 36.4 Multiple Slits (p.1219~1221) 36.5 The Diffraction Grating (p.1221~1225) Interference and diffraction are the phenomena to show the wave nature of light. Interference phenomena occur when two or more waves combine. The colors seen in oil films and soap bubbles are a result of interference between light reflected from the front and back surfaces of a thin film of oil or soap solution. 1. Interference Interference is a phenomenon in which two or more waves superpose to form a resultant wave of greater or lower amplitude. When waves from two sources arrive at point in phase, as in Fig. 1(a), they reinforce each other. The amplitude of the resultant wave is the sum of the amplitudes of the individual waves. This is called constructive interference. Waves from the two sources arrive at point exactly a half-cycle out of phase, as in Fig. 1(b). A crest of one wave meets a trough of the other wave. The resultant amplitude is the difference between the two individual amplitudes. This cancellation is called destructive interference. When light falls on a straightedge and casts a shadow, the edge of the shadow is never perfectly sharp. Some light appears in the area that we expect to be in the shadow, and we find alternating bright and dark fringes in the illuminated area, as in Fig. 4. Such effects are referred to as diffraction. There is no specific, important physical difference between interference and diffraction. However, diffraction commonly means the result from the interference of an infinite number of waves emitted by a continuous distribution of source points. Fig. 1 Conditions for constructive and destructive interference. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 1 / 15

2 Interference effects are most easily seen when we combine waves from monochromatic sources of the same frequency and with a constant phase relationship. We use the term coherent light to refer to the light waves emitted by two such sources. One of the earliest quantitative experiments to reveal the interference of light from two sources was performed in 1800 by Thomas Young. Young s apparatus is shown in perspective in Fig. 2. A light source emits monochromatic light; however, this light is not suitable for use in an interference experiment because emissions from different parts of an ordinary source are not synchronized. To remedy this, the light is directed at a screen with a narrow slit. The light emerging from the slit originated from only a small region of the light source; thus behaves more nearly like the idealized source. (In this experiment, a laser is used as a source of coherent light, and isn t needed.) The light from falls on slits and. Cylindrical wave fronts spread out from and reach and in phase because they travel equal distances from. The waves emerging from and are therefore also always in phase, so and are coherent sources. The interference of waves from and produces a pattern in space. We assume that the distance from the slits to the screen is so large in comparison to the distance between the slits that the lines from and to are very nearly parallel, as in Fig. 2(c). The difference in path length is then given by sin (1) Constructive interference occurs at points where the path difference is an integral number of wavelength,. So the bright regions on the screen in Fig. 2(a) occur at angle for which sin 0, 1, 2, (2) Similarly, destructive interference occurs at points for which the path difference is a half-integral number of wavelengths. sin 1 0, 1, 2, (3) 2 Thus the pattern on the screen is a succession of bright and dark bands parallel to the slits. The intensity at any point can be expressed in terms of the maximum intensity. cos 2 2 sin (3) Fig. 2 (a) Young s experiment to show interference of light passing through two slits. (b) Geometrical analysis of Young s experiment. (c) Approximate geometry when the distance to the screen is much greater than the distance between the slits. Fig. 3 (a) Photograph of interference fringes produced on a screen in Young s double-slit experiment. (b) Intensity distribution in the interference pattern from two identical slits. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 2 / 15

3 2. Diffraction from a single slit Diffraction occurs when light strikes a barrier that has an aperture or an edge. Fig. 4 shows an example of diffraction. The phenomenon of diffraction can be understood using the Huygens s principle, which states that every point on a wave front acts a source of secondary spherical waves. When the waves pass through an aperture, each infinitesimal part of the aperture acts as a source of waves, as shown in Fig. 5(a), and the resulting pattern of light and dark is a result of interference among the waves emanating from these sources. Consider two narrow strips, one just below the top edge of the slit and one at its center, shown in end view in Fig. 6. The difference in path length to is 2sin. If this is equal to 2, cancellation occurs at. Similarly, the light from every strip in the top half cancels out the light from a corresponding strip in the bottom half. Hence a dark fringe occurs whenever 2 sin 2 or sin We may also divide the slit into quarters, sixths, and so on, and use the above argument to show that a dark fringe occurs whenever sin 2,3, and so on. Thus the condition for a dark fringe is (4) sin 1, 2, 3, (5) Fig. 4 An example of diffraction Note that sin 0 0 corresponds to a bright band; in this case, light from the entire slit arrives at in phase. The central bright fringe is wider than the others, as Fig. 7. From Eq. (5) and Fig. 6, we obtain sin. If is much smaller than, is so small that the approximation sin tan is a very good one, and is (6) where is the distance from slit to screen and is the vertical distance of the th dark band from the center. Fig. 5 Diffraction by a single rectangular slit. The long sides of the slit are perpendicular to the figure. Fig. 6 Side view of a horizontal slit. When the distance to the screen is much greater than the slit width, the rays from a distance 2 apart may be considered parallel. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 3 / 15

4 Fig. 7 shows the diffraction pattern of a single slit. The intensity in single-slit diffraction in terms of the angle is sin sin (7) From Eqs. (5) or (7), sin. Thus the width of the central maximum becomes wider when the slit width decreases, as shown in Fig Two Slits of Finite Width In Figs. 2 and 3 we analyzed interference from two very narrow slits and. In this analysis we ignored effects due to the finite (nonzero) slit width. In the more realistic case in which the slits have finite width, the peaks in the two-slit interference pattern are modulated by the single slit diffraction pattern characteristic of the width of each slit. Fig. 9(a) shows the intensity in a single-slit diffraction pattern with slit width. The diffraction minima are labeled by the integer 1, 2,. Fig. 9(b) shows the pattern formed by two very narrow slits with distance 4 between slits. The interference maxima are labeled by the integer 0, 1, 2,. Fig. 7 Intensity versus angle and photograph of the Fraunhofer pattern in single-slit diffraction. Suppose we widen each of the narrow slits forming the pattern of Fig. 9(b) to the same width as that of the single slit in Fig. 9(a). Then the intensities of the two-slit peaks are modulated by the single-slit pattern, which acts as an envelope for the intensity function, as in Fig. 9(c). So the intensity in Fig. 9(c) is proportional to the product of Eqs. (3) and (7): cos 2 sin sin sin (8) Fig. 8 The single-slit diffraction pattern depends on the ratio of the slit width to the wavelength. Fig. 9 Finding the intensity pattern for two slits of finite width. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 4 / 15

5 4. Several Slits and Diffraction Grating Fig. 10 shows an array of several narrow slits, with distance between adjacent slits. Constructive interference occurs for rays at angle to the normal that arrive at point with a path difference between adjacent slits equal to an integer number of wavelengths: sin 0, 1, 2, (9) Equation (9) shows that the maxima in the pattern occur at the same positions as for two slits with the same spacing. However, while there is exactly one intensity minimum between each pair of maxima in the two-slit pattern, there are not the only minima in the multiple-slit pattern, as in Fig. 11. For example, the intensity with eight slits is zero whenever the phase difference is an integer multiple of 4, except when is a multiple of 2 (Fig. 12). Thus there are seven minima for every maximum as in Fig. 11(b). When there are slits, there are 1 minima between each pair of principal maxima. The height of each principal maximum is proportional to and the width of each maximum is proportional to 1. Increasing the number of slits with keeping the spacing of adjacent slits constant gives interference patterns progressively narrower, so their angular position can be measured to very high precision. Fig 10 Multiple-slit diffraction. An array of a large number of slits is called a diffraction grating. When a grating is illuminated by light, the pattern is a series of very sharp lines at angle determined by Eq. (9). So diffraction gratings are widely used to measure the spectrum of light emitted by a source. Fig. 11 Interference patterns for equally spaced, very narrow slits. (a) Two slits. (b) Eight slits. (c) Sixteen slits. The vertical scales are different for each other. is the maximum intensity for a single slit, and the maximum intensity for slits is. The width of each peak is proportional to 1. Fig. 12 Phasor diagrams for light passing through eight narrow slits. Intensity maxima occur when the phase difference 0, 2π, 4π,. Between the maxima at 0 and 2π are seven minima, corresponding to 4, 2, 3 4,, 5 4, 3 2, and Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 5 / 15

6 Equipment 1. List Item(s) Qty. Description PC / Capstone software 1 Records, displays and analyzes the data measured by various sensors. Interface 1 Data acquisition interface designed for use with various sensors, including power supplies which provide up to 15 watts of power. Rotary Motion Sensor 1 Measures angles, angular velocities, etc. of a rotational motion, and using additional accessories, measures position, velocities. etc. of a linear motion. Light Sensor (Post and DIN Cable included) 1 set Measures relative light intensity. Laser sources (Power Adaptor included) 1 set Emits light with a very narrow spectrum. Red : 650 nm Green : 532 nm Optics Bench 1 Provides stable support for holders. Holder Mount (Two Thumbscrews included) 2 sets Provides stable support for optical components. Linear Stage (Thumbscrew & Square Nut, Two Thumbscrews (for fixing Rack) included) 1 Builds a linear positioning mechanism for scanning the light intensity of interference patterns. Rack 1 Converts a linear motion to a rotational motion in combination with the pinion gear inside the Rotary Motion Sensor. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 6 / 15

7 Items Qty. Description Aperture Bracket 1 Acts as a mask and holder for the Light Sensor. Slits 1 Contains several slits. Diffraction Grating 1 set An array of a large number of parallel slits Line spacing: 300 lines/mm, 600 lines/mm Ruler Screen 1 Black screen for observing interference patterns of diffraction gratings. 2. Details (1) Rotary Motion Sensor (3) Aperture Mask / Slits The Rotary Motion Sensor is a bidirectional angle sensor designed to measure rotational or linear position, velocity and acceleration. (Refer to the manual of the magnetic field experiment.) (2) Light Sensor Aperture Mask: narrow 1 6 wide Slits: Slit Width mm / Separation mm Single Slits Variable S.S ~ 0.20 Double Slits 0.04 / / / / 0.50 Variable D.S / ~ 0.75 (4) Diffraction Grating 300 lines/mm, 600 lines/mm The Light Sensor makes measurements of relative light intensity using Si PIN photodiode. It is responsive across a wide spectrum ranging from 320nm through 1100nm. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 7 / 15

8 Setup Setup1. Equipment Setup (3) Use the post to fasten the Light Sensor to the aperture. (1) Mount the Linear Stage on the left end of the optics bench (4) Attach the Light Sensor to the Rotary Motion Sensor. Insert the square nut in the center hole of the Linear Stage into the T-slot located along the center of the Optics Bench. Tighten the thumbscrew to secure the Linear Stage in position. Secure the post of the Light Sensor assembly in the clamp of the Rotary Motion Sensor. Adjust the height of the Light Sensor assembly not to touch the pulley of the Rotary Motion Sensor. (2) Mount the Rotary Motion Sensor on the Rack of the stage. (5) Mount the laser on the right end of the Optics Bench. Insert the Rack into the T-slot on the side of the Rotary Motion Sensor. The teeth on the Rack go through T-slot and then engage the gear that is on the shaft of the Rotary Motion Sensor. Place the Rack with the senor back onto the wide hole of the stage and tighten its thumbscrew. (6) Mount the slit disk. Attach the slit disk to the holder and mount the slit assembly right in front of the laser with the printed side toward the laser. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 8 / 15

9 (7) Plug the sensors into the inputs of the interface. (11) Select a slit. Slit Width mm / Separation mm Single Slits Variable S.S ~ 0.20 Double Slits 0.04 / / / / 0.50 Variable D.S / ~ 0.75 (12) Align the laser beam. (9) Set the sensitivity of the Light Sensor. Set the Light Sensor for maximum sensitivity by sliding [GAIN] to 100 0~5 lux. If the measured light intensity goes too high (flat line at 100% on the graph), adjust it to 10 0~50 lux or 1 0~500 lux. Note that lowering the sensitivity makes the resolution of readings worse. Move the sensor assembly until you can see the beam on the aperture mask. Use the adjustment screws on the laser to adjust the position of laser beam to pass the slit and to make the pattern on the aperture as bright as possible. (10) Select an aperture. Caution It is recommended to use aperture #2. If required, you can use any other aperture. If you use a narrower aperture, it is difficult to observe the light with low intensity. If you use a wider aperture, measured light intensity goes too high. The laser beam is so concentrated that it can cause real damage to your retina if you look into the beam either directly or by reflection from a shiny object. Do NOT shine them at others or yourself. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 9 / 15

10 Setup2. Software (Capstone) Setup Select [Rack & Pinion] for [Linear Accessory]. (1) Add sensors. The interface automatically recognizes the Rotary Motion Sensor. [Change Sign] switches the sign on the sensor. The sign of collected data depends on the setup status or the rotational direction of the shaft. Activate [Change Sign] if required. Click the input port which you plugged the Light Sensor into and select [Light Sensor] (yellow icon) from the list. (3) Adjust the sample rate of measurements. Select [20.00 Hz] for all sensors, or other value if required. (4) Create a graph display. (2) Configure the Rotary Motion Sensor. Click the Rotary Motion Sensor icon and then click the properties button ( ). Click and drag the [Graph] icon from the [Displays] palette into the workbook page. Select [Position(m)] of the Rotary Motion Sensor for the -axis and [Light Intensity(%)] of the Light Sensor for the -axis. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 10 / 15

11 Procedure Caution The laser beam is so concentrated that it can cause real damage to your retina if you look into the beam either directly or by reflection from a shiny object. Do NOT shine them at others or yourself. (6) Scale the graph. Adjust the scale of the graph by clicking [Scale axes ] or by dragging the mouse pointer on the graph. Experiment 1. Diffraction from a Single Slit (1) Set up your equipment. (See Setup 1.) (2) Set up the Capstone program. (See Setup 2.) (3) Select a slit and record the width of the slit. m (4) Measure the distance between the slit (front side of slit disk) and the aperture mask. m (5) Scan the interference pattern. Click [Record]. Then slowly turn the pulley of the Rotary Motion Sensor to scan the pattern. Hold the rear of the Rotary Motion Sensor down against the linear stage bracket so it does not wobble up and down as it moves. Note If measured light intensity goes too high (it will flat line at 100% on the graph), 1 Place the slit and the sensor assembly as far away as possible. 2 Choose a narrower aperture. (Note that you may not be able to measure the light of low intensity if the aperture is too narrow.) 3 Change [GAIN] switch on the Light Sensor from 100 0~5 lux to 10 0~50 lux. (In this case, the resolution of readings becomes worse.) Click [Stop] when you have finished the scan. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 11 / 15

12 (7) Analyze the graph. (8) Record your data and analyze the result. Click [Show coordinates ] to read off data points. Slit Pattern m min m sin sin average 650nmred 532nmgreen reference (9) Repeat the steps (5)-(8) for the other single-slits. (10) Repeat the steps (3)-(9) for the other laser source. (11) Repeat for the variable single slit. Note Using your eyes, make an observation of changes in pattern as you decrease the slit width. You can modify the properties of the graph if required. Q How does the single slit pattern change as you decrease the slit width? What is expected if the slit width decreases to the light wavelength level? (Refer to Eqs. (5), (7) and Fig. 8.) A Note The manufacturer of the slits claims the uncertainty in the slit width is 0.005mm. This means the % uncertainty in using the 0.02mm slit width is 20% and for the 0.16mm slit width is 3%. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 12 / 15

13 Experiment 2. Diffraction from Two Slits Repeat Experiment 1 for double-slits. Q Compare the patterns of different slit widths (same slit separations) and explain why. A Q What is expected if the slit width decreases to the light wavelength level? A Slit Pattern m max m sin average sin Q Compare the patterns of different slit separations (same slit widths). Also, when you use the variable double-slit, how does the pattern change as you decrease the slit separation? Describe the changes and explain why. 650nmred 532nmgreen reference A Q Your graph may be different from Fig. 3. Explain why. Note A The manufacturer of the slits claims the uncertainty in the slit separation is 0.01mm. This means the % uncertainty in using the 0.25mm slit separation is 4% and for the 0.50mm slit separation is 2%. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 13 / 15

14 Experiment 3. Diffraction Grating (1) Set up your equipment. (4) Measure the distance from the central maximum to the first maximum. Mount the laser, the diffraction grating, and the ruler screen on the optics bench. Note If you cannot find the first maximum on the screen, check the following. 1 If the screen is too far from the diffraction grating, the first maximum reaches out of the screen. Place the screen near the grating. 2 When the angle is too large, the first maximum reaches inside surface of the holder. Attach the diffraction grating on the screen-facing side of the holder. (5) Record your data and analyze the result. Caution separation grating screen m 1 max m sin sin Handle the diffraction gratings with care. They are very fragile. 650nmred 532nmgreen reference (2) Calculate the slit (line) separation of the diffraction grating. (6) Repeat the steps (2)-(5) for the other diffraction grating. m (3) Measure the distance between the grating and the screen. (7) Repeat the steps (2)-(6) for the other laser source. m 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 14 / 15

15 General Physics Lab (International Campus) Result & Discussion Your TA will inform you of the guidelines for writing the laboratory report during the lecture. End of LAB Checklist Please put your equipment in order as shown below. Delete your data files and empty the trash can from the lab computer. Turn off the Computer and the Interface. Unplug the dc Adaptor of the laser source. Handle the Diffraction Gratings with care. They are very fragile. Tighten all Thumbscrews and Nuts in position. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 15 / 15