Part I: Measuring the Wavelength of Light

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2 In other words, the additional bright spots one sees on the wall are due to reflections that are only possible because we are preventing them from being canceled by destructive interference. We secure more reflections by arranging for fewer possibilities of reflection!! Continuing with this line of reasoning thus suggests that the bright spots are due to constructive interference. Figure 1, which is not to scale, shows the experimental arrangement. The ruler is placed on a table about 2 m (distance L) from the wall, and the laser is positioned so that the beam just strikes near the end of the ruler at a grazing angle φ 0 (Note that φ 0 is complementary to the angle to the normal, θ 0 ). Part of the laser beam misses the ruler completely and continues undeviated to the wall ( direct beam ). Many reflections will appear on the wall, but, to keep the drawing simple, only two are shown in the figure. The brightest reflected spot, the central bright spot, corresponds to the reflection whose angle is equal to the angle, φ 0. Many more reflection spots will be present, above and below φ 0. L first spot above center laser φ 0 θ 0 ruler φ 0 φ 0 φ 1 central reflected bright spot y 0 y 1 tan φ = y / L and tan φ 1 = y1 / L 0 0 direct beam Figure 1 wall You may recall that the bright spots from a diffraction grating occur at angles such that sin θn = nλ / d (1) but this equation describes light arriving perpendicular to the surface ( θ 0 = 0 ). In this lab you have light arriving at an angle to the normal θ0 0, and for this more general case, constructive interference (bright spots) occur according to 2

3 sinθ sin θ = λ /, where n = 0,1,2,.... (2) n 0 n d [The absolute value is used to ensure a positive value; the sign would otherwise depend on angle conventions and upon whether the light is being reflected from small stripes (as in this case) or transmitted through small openings. In this equation, both n and the angles are positive values.] Figure 2 shows a more detailed illustration of the light hitting the ruler in our specific geometry. rays from incident laser beam rays to first spot (above central spot) non-reflective black markings ruler edge φ 0 φ 1 d d Figure 2 d = 1/64" = mm The angles φ0, φ1, φ2, etc. can be determined fairly easily by measurement of distances (see Figure 1), but their complements θ0, θ1, θ 2, etc. are more difficult to determine directly. Thus we will use the complementary relationship to rewrite the angle condition as nλ / d = cosφ cosφ (3) n 0 When λ << d, as it is in this experiment, cosφn cosφ0 must be extremely small for equation (3) to be true if n is a small integer (= 1, 2, or 3). However, due to the way in which cosine varies with the angle for small angles, φ 0 and φ n can be small yet large enough for reasonably easy measurement, while cosφn cosφ0 remains very small. This is the secret why the grazing method will measure wavelengths that are very short compared to the relatively large spacing on the ruler. Note that all angles in the diagram are shown greatly exaggerated. 3

4 Measurement procedure (a) Allow the ruler to slightly overhang the edge of the table then adjust the angle of the laser beam. The grazing angle should be < 2, and the ruler must be perpendicular to the wall. Tape a piece of paper on the wall for viewing and marking the positions of the various spots. Note : The laser can be kept turned on using a clothes pin, but please turn off the laser when not performing measurements! Notice that on your ruler there are a series of marks with different spacings and lengths interspersed. Make sure you are using only the 1/64 marks. (b) Verify which set of marks are producing the spots you see on the wall by sliding the ruler to the left and right and see sets of spots appear and disappear. (c) Record as many spots as you can. The angles would be difficult to measure directly (with a protractor, for example), but the geometry indicated in Figure 1 shows you how to determine the angles by distance measurements. Hint: The distance from the central bright spot to the place where the direct beam strikes the wall is 2 y 0 (see Figure 1). (d) Make measurements as needed to determine φ 0, φ 1, φ 2, and φ 3, then use Equation (3) to determine the wavelength of the laser light. Also estimate the uncertainty in your value for the wavelength. Part II: Determining the Amount of Data on a CD Now that you have measured the wavelength of the laser using a ruler, you can use the laser as a ruler to measure the spacing between tracks on a compact disc (CD)! From this, you may determine the maximum amount of information that can be stored on a CD. CD diffraction The surface of a CD is a highly reflective layer containing a spiral path of small marks, or pits. If stretched out, this spiral would be about 5 km long! The digital data are stored in a code, according to the pit length and the distance between pits. [Imagine an advanced style of Morse Code: dot-dash etc.] The pits are arranged along the spiral path in tracks, as shown in Figure 3. A portion of one track is highlighted by the white box. The depth of each pit is 0.11 µm and the width is 0.5 µm. Pit length and spacing varies, as can be seen, but the average spacing from pit to pit is roughly the same size as the distance between the tracks of pits. 4

5 data recording area Photograph of pits on a CD. From On the Surface of Things, by Felice Frankel and George Whitesides. Figure 3 The tracks of pits and the unbroken stripes between them behave in much the same way as a reflective diffraction grating, but here the stripes curve gradually around the CD within the recording area (Figure 3). The fine stripes are why you see beautiful rainbow colors when white light illuminates the CD. However, when light of one wavelength (here, a laser beam) is reflected off the disc, a diffraction pattern is formed. If the angle of incidence is close to the normal, the condition for constructive interference is identical to that for a transmission diffraction grating (see Equation 1). In the present example, the rows of pits (tracks) make the grating, and the distance between the rows of pits, d, can be determined from Equation 1. By measuring θ and using your value of λ from Part I, you can use Equation (1) to calculate the distance between rows of pits. The experimental arrangement is illustrated in Figure 4. Measurement procedure (a) Arrange a disc, laser, and screen in a manner similar to Figure 4a. Direct the laser beam such that it strikes the CD approximately half way up, as shown in Figure 4b. Adjust the angle between the laser beam and the CD until the direct mirror reflection (n = 0) comes back to hit the front of the laser pointer (you may need to have it hit 2-3mm above where it comes out, in order to see it). Doing this will insure that the incident laser beam is normal to the surface of the CD. You should be able to see the first-order maximum (bright spot) on the screen. 5

6 (b) Measure distances as needed to determine the angle of the first-order maximum (bright spot), then calculate the spacing d from Equation (1). laser laser beam CD CD screen Figure 4a 4b Information stored on a CD Now calculate the maximum amount of information (number of pits or, literally, bits of information) that can be put on a CD. (c) First, calculate the area occupied by one bit, using the simple model that it occupies a rectangle of width d and length l. To simplify things, assume that, on average, l is equal to d. (d) Now calculate the area of the CD that is actually recorded with data. You will need to measure the radii of the circles defining the recorded area on your CD, and remember that the area of a circular band is the difference of the areas of the large and small circles. The total number of bits that can be recorded is then the total recorded area divided by the area for each bit. How many bits can be stored on the CD? Data amounts are more often given in bytes, where 1byte = 8 bits. Also, 1 MB = 1 megabyte 6 = 10 bytes. Based on your simple model and your measurements, how many megabytes can be stored on the CD? The actual amount of storage on a CD is about 650 to 700MB. 6

7 Additional comments You may be interested to know that the table of contents for the whole CD is recorded on the lead-in portion consisting of about 30 tracks (one track equals one trip around the disc). The width of this program area corresponds to the thickness of one human hair. To detect individual pits, the laser beam must be focused down to a spot about 1µm in diameter. This should give you an idea of how remarkably spatially coherent the light is and an appreciation for the technological obstacles that must be overcome to track these bits of information. The CD player is literally operating at the diffraction limit (resolution) of light! And now we have DVD discs! Check the table below for a comparison between DVD and CD discs. Specifications CD DVD Disc Diameter 120 mm 120 mm Disc thickness 1.2 mm 1.2 mm Disc structure Single substrate Two bonded 0.6mm substrates Laser wavelength 780 nm (infrared) 650 and 635 nm (red) Numerical aperture Track pitch 1.6 µm 0.74 µm Shortest pit/land length.83 µm 0.4 µm Data layers 1 1 or 2 Data capacity 680 megabytes Single layer 4.7 gigabytes Dual layer 8.5 gigabytes Data rate Data rate 165 kilobytes/sec Data rate 1,000 kilobytes/sec 7

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