Exercise 8: Interference and diffraction

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1 Physics 223 Name: Exercise 8: Interference and diffraction 1. In a two-slit Young s interference experiment, the aperture (the mask with the two slits) to screen distance is 2.0 m, and a red light of wavelength 600. nm is used as the light source. If you want a bright fringe spacing of 1.0 mm, what is the required slit separation? 2. If a piece of glass (n = 1.5) is coated with a transparent plastic (n = 2.0), what thickness or thicknesses give the maximum reflection? What thickness or thicknesses give the minimum reflection? Assume that a light of wavelength 500. nm is used.

2 3. You move one mirror of a Michelson interferometer (page 1177, figure 35.19) through a distance of m, and see exactly 850 bright fringes pass by. What is the wavelength of illumination (in nm)? 4. a. A Michelson interferometer may be used to determine the index of refraction of a gas by placing an evacuated cell along the entire length of the arm with the movable mirror (L2 in figure 35.19), and gas is slowly added to the cell. The interference fringes are counted as they move across the view aperture when the gas flows into the cell. Show that the effective optical path difference of the light beam for the cell full of gas versus the evacuated cell is 2L2 (n 1), where n is the index of refraction of the gas. Hint: set up equations for the effective optical path lengths of the evacuated cell and the cell full of gas of refractive index n, then subtract.

3 b. For the setup in part a, show that the total number of fringes (both light and dark) that move across the field of view of the aperture is 4L2 (n 1)/l, where l is wavelength of illumination. c. How many total fringes would be counted if air (n = ) was introduced into the evacuated cell (L2 = cm), using yellow sodium light of nm? 5. Laser light (l = 633 nm) is shined upon a single slit of width 0.25 mm. The observation screen is 2.0 m from the slit. What is the width of the central bright fringe? What is the width of the bright fringe between the 5 th and 6 th minima (dark fringes)?

4 6. For a potential experiment on diffraction, you are given a diffraction grating that has 5000 slits (exactly) per centimeter. The experiment will determine the separation of the second-order red (l = nm) and blue (l = nm) light fringes on a screen 1.00 m away from the grating. a. Determine the distance between the slit centers. b. Determine the angular deviations q (angle of the line from center of slit to the m th dark fringe on the screen) for the second order red and blue light. c. Determine the separation distance (in m) on the screen between the second order red and blue fringes.

5 7. a. Determine the size of the Airy disk (in m) found at the center of a 4.00-cm diameter lens, with a focal length of 15.0 cm. Assume the incident light wavelength is the middle of the visible spectrum = 550. nm. b. In observational astronomy, we assume that stars, being so far away, are point sources of light, and that the image of a star in a telescope eyepiece is therefore also a point. Given that the average human near-field resolution is 0.10 mm, does your result in part a justify this assumption? Explain your answer, using the value from part a.

6 c. Assume that the objective lens diffraction limit is the only one that matters on a telescope (actually a good assumption, not justified here). What is the angular size (in radians) of the smallest object that can be truly observed as a disk on the 4.00-cm telescope in part a? Can Jupiter (maximum angular size = 51 arc-seconds) be seen as a disk through this telescope? Note that real telescopes have glass or mirror imperfections which makes even achieving the diffraction limit a challenge.

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