66 #24 Interference of Light Waves Objective The objective of this experiment is to observe various manifestations of the interfer- ence of light waves. Introduction and Theory Because light has a wave nature it can exhibit the same phenomena that, say, water waves do when they interact with sharp boundaries or with other waves. The effects of interaction include reflection, refraction, diffraction, and interference. The effects of dif- fraction and interference will be the subject of this laboratory. I. The Diffraction Grating The diffraction grating consists of a very large number of fine, equally spaced paral- lel slits. There are two types of diffraction gratings: the reflecting type and the transmit- ting type. The lines of the reflection grating are ruled on a polished metal surface: the incident light is reflected from the unruled portions. The lines of the transmission grating are rule on glass: the unruled portions of the glass act as slits. Gratings have typically be- tween 100 to 800 lines per millimeter. A diffraction grating provides the simplest and most accurate method for measuring wavelengths of light. The principles of diffraction and interference are applied to the measurement of wavelengths with a diffraction grating. Let the vertical broken line in Fig. 24.1 represent a magnified portion of a diffraction grating. Let a beam of parallel monochromatic light, originally from a single source and having passed through a slit, impinge upon the grating from the left. By Huygens' principle, the light spreads out in every direction from the ap- ertures of the grating, each of which acts as a separate new source of light. The envelope of the secondary wavelets determines the position of the advancing wave. In Fig. 24.1 we see the instantaneous positions of several successive wavelets after they have advanced beyond the grating. Lines drawn tangent to these wavelets connect points which are in phase: hence they represent the new wave fronts. One of these wave fronts is tangent to wavelets which have all advanced the same distance from the slits, and the wave front formed is thus parallel to the original wave front. A converging lens placed in the path of these rays would form the central image. Another wave front is tangent to wavelets whose distances from adjacent slits differ by one wavelength. This wave front advances in the direction 1 and forms the first-order image. The next wave front is tangent to wavelets whose distances from adjacent slits differ by two wavelengths. This wave front advances in the direction 2 and forms the second-order image. Images of higher orders will be found at correspondingly greater angles. 67 Max Max Max Incident Max Wave S2 Max Max Max THEN So Max Max Max Si Max Max Max с B Fig. 24.1. Interference patterns resulting from waves incident on two slits of a grating Let be the wavelength of the light, m the order of the image (m= 1 for the first- order image, m = 2 for the second-order image, etc.) formed on either side of the central (m= 0) image, d the distance between slits (a diffraction grating consists of several equal- ly-spaced slits), and the angle of deviation from the original direction of the light. These four quantities are interrelated by the grating equation, m2 = d sin e (24.1) The angle O is measured directly, while the grating constant d is calculated from the number of lines per millimeter ruled on the grating. If the light is monochromatic, a single image of the slit will appear in each order. - Grating 12 11 -Source ө Y Slit 11 Observer 12 Fig. 24.2 Viewing diffraction-produced (virtual) images of a slit through a grating. 68 If the light is polychromatic, however, there will be as many images of the slit in each order as there are different wavelengths in the light from the source, the diffracting angle for each wavelength being given by the above equation. The resulting pattern of multiple lines in the first order is known as the first-order spectrum, that in the second order is the second-order spectrum, and so on. Spectra produced in this manner will be discussed in detail in Experiment #24. ni 12 13 11 t Fig. 24.3. Interference of two beams incident on the left and right boundaries of a thin film. II. Thin-Film Interference Interference is the combining by superposition of two or more waves that meet at one point in space. Thin-film interference can be understood as the combining by super- position of rays reflected from opposite sides of the thin film. Examples of thin films are soap films, lens coatings, or a thin wedge of air formed between two glass plates as shown in Fig. 24.3. Ray rı is reflected from the left surface of the film and ray r2 is reflected from the right surface. If t is the thickness of the air wedge at the point where ray rı strikes it, then ray r2 will travel a distance 2t further than ray rı. Therefore, if the rays were in phase be- fore they reached the wedge, when they combine they will no longer be in phase. Three effects must be taken into account in determining the net result of this type of superposition. First, not only does a ray undergo a partial reflection when encountering another medium (glass-to-air or air-to-glass), but it may also experience a 180° phase shift (which corresponds to a shift of 1/2 in the wavetrain). A 180° phase shift occurs whenever the index of refraction of the first medium is less than that of the second (e.g., air-to- glass). Such a change of phase does not occur, however, when the index of refraction of the first medium is greater than that of the second (e.g., glass-to-air). Because of this ef- fect, the two rays will travel essentially the same distance (assuming t«< 2 in Fig. 24.3), but ray T2 will experience a 180° phase shift upon reflection while ray rı will experience 69 no phase shift at all. Therefore, the two rays will emerge from the film 180° out of phase and subsequently combine destructively. This region of the film will appear dark to an observer's eye. The second effect involves the fact that a depends on the index of refraction, n. If a is the wavelength in a vacuum, the corresponding wavelength in a medium of index of refraction n, an, is given by a an (24.2) n This is important because the extra distance traveled by one of the two rays is al- ways traveled in the thin film, so it is in that should be used to determine whether or not there is constructive (bright) or destructive (dark) interference. For the situation depicted in Fig. 24.3, 2n = 2 because the index of refraction of air is 1.00. The third effect is a path difference which arises between the two rays when the film thickness is non-negligible. In the discussion of the phase shift given above, we assumed for simplicity that t«< 1, so that the two rays would always emerge from the film 180° out of phase. If, however, the distance traveled by ray r2 in the film is an appreciable frac- tion of the wave-length, then this ray will be effectively shifted back (relative to ray rı) by an amount equal to twice the thickness of the air film. For example, if 2t = 122, then ray r2 will be shifted back by exactly half of one wavelength; this effectively cancels the 180° phase shift, thus yielding constructive interference (bright). Similarly, constructive inter- -21 ference would result for 2t = ža, z ł, ż 1, etc. If, on the other hand, 2t = 1, then ray 12 will be shifted back by exactly one wavelength, so that the 180° phase shift yields destruc- tive interference (just as in the case where t was negligible compared with 2). Similarly, destructive interference will occur when 2t = 27, 37, 41, 52, etc. In general, the, the condition for destructive interference at ap- proximately normal incidence of light is 3 5 2t = min (m= 0, 1, 2, 3, ...) (24.3) where 2t (twice the film thickness) is the path difference between the two rays, m is a pos- itive integer, and in is the wavelength of light in the film. When a wedge of film is used the film thickness varies from essentially zero near the apex of the wedge to a maximum thickness t. To illustrate the effects described above, suppose that light of wavelength 1 = 2n = 500 nm illuminates at approximately normal incidence a film of thickness t = 0.01 nm = 10 um = 10,000 nm. Then the extra distance traveled by ray r2 (i.e., the path difference between the rays) is 2t = 20,000 nm = 20,000 nm/500 nm = 40 wave- lengths, so there will be 40 changes (interference fringes) from dark (destructive) at the apex of the wedge to bright (constructive) where the thickness is t. More importantly, if the number of interference fringes can be counted between two points on the film, then the difference in thickness between those two points can be calculated, and so this technique can be used to measure very tiny dimensions such as the diameter of a human hair. Apparatus - The following equipment will be needed: monochromatic light source (1 = 546.1 nm) mercury vapor discharge tube with high-voltage power supply optical bench ruled T-bar with slit 100, 300, and 600 line/mm diffraction gratings optical flat glass microscope slide - - Experimental Procedure I. Diffraction Grating The arrangement used in this experiment is shown in Fig. 24.2. Light from a mono- chromatic source passes through a narrow slit and falls on the grating. The observer looks through the grating in the direction of the light source and sees the central (zeroth-order) image and the diffracted beams which make up the successively higher-order images (first, second, etc.) on either side of the central image. The measuring scale should be mounted horizontally and perpendicular to the central ray. The grating should be mounted parallel to the scale about 50 cm away from it and with the grating rulings parallel to the slit. 1. Mount the 100 lines/mm grating and slit with the scale on the optical bench, with the grating rulings parallel to the slit and the plane of the grating and the slit with the scale both perpendicular to the central ray. 2. Place the monochromatic light source in position behind the slit. Look through the dif- fraction grating in the direction of the source. Keep your eye several inches away from the grating. As discussed above you will see the source and virtual images of the source on either side. Carefully move the grating holder forward toward the slit until the first-, second-, and third-order slit images are positioned clearly on the scale on each side of the central (zeroth-order) image. A slight adjustment can perhaps be made to render both the left-hand and right-hand image distances equal. Measure and record the dis- tance between the slit and the grating. do not change the position of the grating until the next grating is used. 3. Look through the grating at the first-, second-, and third-order (virtual) images on the right-hand side. Record the apparent position and each image on the T-bar scale. Re- peat for the left-hand side. 4. Repeat steps 1-3 for the 300 lines/mm grating. 5. Repeat steps 1-3 for the 600 lines/mm grating. (Note: You will only be able to observe the first two orders.)
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