FUNDAMENTALS OF PHOTONICS Mandoli, D. F., and W. R. Briggs. “Fiber Optics in Plants,” Scientific American (August 1984): 90. Wehner, R. “Polarized Light Navigation by Insects,” Scientific American (July 1976): 106. Weisskopf, Victor F. “How Light Interacts with Matter,” Lasers and Light, Readings from Scientific American. W. H. Freeman and Company, 1969, pp 14-26. Optical Manufacturing Companies Optics and Optical Instruments Catalog. Edmund Scientific, Industrial Optics Division, Barrington, New Jersey. Melles-Griot Catalog. Melles-Griot Optical Systems, Rochester, New York, and MellesGriot Photonics Components, Irvine, California. Problem Exercises 1. A HeNe laser in air emits light at 632.8 nanometers. The beam passes into a glass substance of index 1.35. What are the speed, wavelength, and frequency of the laser light in the glass substance? 2. Use the principle of superposition to sketch a picture of the resultant waveform for the two waves shown at the right. Does the resultant wave ever reach an amplitude of 2A? 3. A Young’s double-slit interference experiment is carried out with blue-green argon laser light. The interference pattern formed on a screen 3.3 m away contains the first-order (m = 1) bright fringe 3.4 mm from the center of the pattern. If the separation of the double slits is 0.50 mm, what is the wavelength of the argon laser light? 164 BASIC PHYSICAL OPTICS 4. Suppose you are asked to design a nonreflecting surface for a Stealth aircraft. Your goal is to select an antireflective polymer of index 1.6 and of optimum thickness, so that radar waves of wavelength λ = 3.5 cm will not be reflected from the aircraft surface. What is the thinnest layer of polymer you can apply to achieve your goal? 5. Solar cells made of silicon (Si) are designed with nonreflecting, thin-film coatings such as silicon monoxide (SiO) to minimize the reflection of incident light. Determine the thickness of the thinnest film of SiO that will cause the least reflection of sunlight. Take the average wavelength of sunlight to be near 550 nanometers. 6. An oil spill on an ocean coastline near you produces an oil slick on the water for miles around. After a few days, you take a helicopter ride out over the water and notice—with the help of a handheld spectrometer—that the oil slick produces a first-order maximum of reflected light of wavelength 550 nanometers. What is the thickness of the oil slick at that time? Assume the oil has n = 1.25 and saltwater has n = 1.34. 7. A laser beam of unknown wavelength is incident on a single slit of width 0.25 mm and forms a Fraunhofer diffraction pattern on a screen 2.0 m away. The width of the central bright fringe is measured to be about 7 mm. What is the wavelength of the laser light? What might the laser be? 8. A thin layer of liquid (methylene iodide) is sandwiched between two glass microscope slides, as shown in the accompanying sketch. Light of wavelength λ = 550 nm is incident on the glassliquid interface. The liquid has an index of refraction of n = 1.76. (a) Is there an automatic phase shift of λ/2 for the light reflected at the top face of the liquid film? (b) Is there an automatic phase shift of λ/2 for the light reflected at the bottom of the film? (c) What is the minimum thickness of liquid film required if the light incident perpendicularly on the sandwich is to be strongly reflected? 165 FUNDAMENTALS OF PHOTONICS 9. Nearly plane waves of C02 laser light of wavelength λ = 10.6 µm emerge from a circular aperture 10 cm in diameter. If the C02 laser light is to be examined on a target in the far field, about how far from the aperture should this examination take place? 10. Refer to Figure 4-20, which shows a Fraunhofer diffraction pattern for a circular aperture. If the aperture is of diameter 150 µm and a helium-cadmium laser of wavelength λ = 442 nm is used to illuminate the aperture, determine (a) an approximate far-field distance from aperture to screen for a Fraunhofer pattern, (b) the half-angle beam divergence of the laser beam in the far field, and (c) the radius of the airy disk on the screen. 11. If one were to send a laser beam of wavelength 694 nm through a telescope aperture of diameter 2.5 meters on toward the moon, 3.84 × 105 km away, what would be the diameter of the laser beam when it hit the moon’s surface? 12. What is the angular separation in second order between light of wavelength 400 nm and 700 nm when the light is incident normally (perpendicularly) on a diffraction grating of 5000 grooves/cm? 13. Vertically polarized light of intensity I0 is incident on a polarizer whose TA makes an angle of 30° with the vertical. The light then passes through a second polarizer whose TA makes an angle of 90° with the vertical, as shown in the sketch. (a) What is the intensity of the light, in terms of I0 , that passes through the second polarizer? (b) What is its orientation? 166 BASIC PHYSICAL OPTICS 14. A submarine floating on the ocean (n = 1.34) transmits a message to a receiver on a 100-m high cliff located at the coastline. The submarine’s antenna extends 5.5 m above the ocean surface. (a) If the transmitted signal is completely polarized by reflection from the ocean surface, how far must the submarine’s antenna be from the coastline? (b) Relative to the ocean surface, what is the direction of polarization of the polarized signal? 15. Figure 4-30 shows the two Brewster windows tilted toward one another, each at the appropriate angle with the horizontal. See sketch (a). What would be gained or lost by having the windows parallel to each other, as in sketch (b)? Hint: Use your knowledge of Snell’s law to draw a center ray completely through both windows from M1 to M2. Use Figure 4-31 to help with details at each window. What happens to the center ray for each sketch? 167 16. White light contains all wavelengths from deep blue at 400 nm to deep red at 700 nm. A narrow beam of collimated white light is sent through a prism of apex angle 20° as shown. The prism is made of light flint glass whose refractive index at 400 nm is 1.60 and at 700 nm is 1.565. What is the angular spread between the red and blue light at the minimum angle of deviation for each? 17. What is the light cone acceptance angle for an optical fiber of diameter 100 µ, located in air, having a plastic core of index 1.49 and a plastic cladding of index 1.39? Make a sketch of the fiber, showing a limiting ray along the surface of the acceptance cone entering the fiber and refracting appropriately. 18. A laser beam is incident on the end face of a cylindrical rod of material as shown in the sketch. The refractive index of the rod is 1.49. How many internal reflections does the laser beam experience before it exits the rod?