Chapter #10 Solutions - Optics - Eugene Hecht - 5th Edition

1. A point source S is a perpendicular distance R away from the center of a circular hole of radius a in an opaque screen. If the distance to the periphery is (R + ℓ), show that Fraunhofer diffraction will occur on a very distant screen whenλR» a2/2What is the smallest satisfactory value of R if the hole has a radius of 1 mm, ℓ≤λ/10, and λ - 500 nm? Get solution

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3. Referring back to the multiple antenna system on p.451, compute the angular separation between successive lobes or principal maxima and the width of the central maximum. Get solution

4. Examine the setup of Fig. 10.3 in order to determine what is happening in the image space of the lenses; in other words, locate the exit pupil and relate it to the diffraction process. Show that the configurations in Fig. P. 10.5 are equivalent to those of Fig. 10.3 and will therefore result in Fraunhofer diffraction. Design at least one more such arrangement.Figure P.10.5... Get solution

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6. The angular distance between the center and the first minimum of a single-slit Fraunhofer diffraction pattern is called the half-angular breadth; write an expression for it. Find the corresponding half-linear width (a) when no focusing lens is present and the slit-viewing screen distance is L, and (b) when a lens of focal length f2 is very close to the aperture, Notice that the half-linear width is also the distance between the successive minima. Get solution

7. A single slit in an opaque screen 0.10 mm wide is illuminated (in air) by plane waves from a krypton ion laser (λ0 — 461.9 nm). If the observing screen is 1.0 m away, determine whether or not the resulting diffraction pattern will be of the far-field variety and then compute the angular width of the central maximum. Get solution

8. A narrow single slit (in air) in an opaque screen is illuminated by infrared from a He-Ne laser at 1152.2 nm, and it is found that the center of the tenth dark band in the Fraunhofer pattern lies at an angle of 6.2° off the central axis. Please detennine the width of the slit. At what angle will the tenth minimum appear if the entire arrangement is immersed in water (nw, = 1.33) rather than air (na = 1.00029)? Get solution

9. A collimated beam of microwaves impinges on a metal screen that contains a long horizontal slit that is 20 cm wide. A detector moving parallel to the screen in the far-field region locates the first minimum of irradiance at an angle of 36.87° above the central axis. Determine the wavelength of the radiation. Get solution

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14. Show that for a double-slit Fraunhofer pattern, if a = mb, the number of bright fringes (or parts thereof) within the central diffraction maximum will be equal to 2m. Get solution

15. Two long slits 0.10 mm wide, separated by 0.20 mm, in an opaque screen are illuminated by light with a wavelength of 500 nm. If the plane of observation is 2.5 m away, will the pattern correspond to Fraunhofer or Fresnel diffraction? How many Young's fringes will be seen within the central bright band? Get solution

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17. What is the relative irradiance of the subsidiary maxima in a three-slit Fraunhofer diffraction pattern? Draw a graph of the irradiance distribution, when a = 2b, for two and then three slits. Get solution

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22. Starting with the irradiance expression for a finite slit, shrink, the slit down to a minuscule area element and show that it emits equally in all directions. Get solution

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25. Show that Fraunhofer diffraction patterns have a center of symmetry [i.e., I(Y, Z) = I( -Y, -Z)], regardless of the configuration of the aperture, as long as there are no phase variations in the field over the region of the hole. Begin with Eq. (10.41). We'll see later (Chapter 11) that this restriction is equivalent to saying that the aperture function is real. Get solution

26. With the results of Problem 10.14 in mind, discuss the symmetries that would be evident in the Fraunhofer diffraction pattern of an aperture that is itself symmetrical about a line (assuming normally incident quasimonochromatic plane waves). Get solution

27. From symmetry considerations, create a rough sketch of the Fraunhofer diffraction patterns of an equilateral triangular aperture and an aperture in the form of a plus sign. Get solution

28. Figure P. 10.17 is the irradiance distribution in the far field for a configuration of elongated rectangular apertures. Describe the arrangement of holes that would give rise to such a pattern and give your reasoning in detail.Figure P.10.17 (Photo courtesy R. G. Wilson, Illinois Wesleyan University.)... Get solution

29. In Fig. P.10.18a and b arc the electric field and irradiance distributions, respectively, in the far field for a configuration of elongated rectangular apertures. Describe the arrangement of holes that would give rise to such patterns and discuss your reasoning.Figure P.10.18 (Photo courtesy R. G. Wilson, Illinois Wesleyan University.)... Get solution

30. Figure P. 10.19 is a computer-generated Fraunhofer irradiance distribution. Describe the aperture that would give rise to such a pattern and give your reasoning in detail.Figure P.10.19 (Photo courtesy R. G. Wilson, Illinois Wesleyan University.)... Get solution

31. Figure P. 10.20 is the electric-field distribution in the far field for a hole of some sort in an opaque screen. Describe the aperture that would give rise to such a pattern and give your reasoning in detail.Figure P.10.20 (Photo courtesy R. G. Wilson, Illinois Wesleyan University.)... Get solution

32. In light of the five previous questions, identify Fig. P.10.21, explaining what it is and what aperture gave rise to it.Figure P.10.21 (Photo courtesy R. G. Wilson, Illinois Wesleyan University.)... Get solution

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36. Verify that the peak irradiance I1 of the first "ring" in the Airy pattern for far-field diffraction at a circular aperture is such that I1/I(0) = 0.0175. You might want to use the fact that... Get solution

38. No lens can focus light down to a perfect point because there will always be some diffraction. Estimate the size of the minimum spot of light that can be expected al the focus of a lens. Discuss the relationship among the focal length, the lens diameter, and the spot size. Take the f-number of the lens to be roughly 0.8 or 0.9, which is just about what you can expect for a fast lens. Get solution

39. Figure P. 10.24 shows several aperture configurations. Roughly sketch the Fraunhofer patterns for each. Note that the circular regions should generate Airy-like ring systems centered at the origin.Figure P.10.24... Get solution

40. Suppose that we have a laser emitting a diffraction-limited beam (λ0 = 632.84 nm) with a 2-mm diameter. How big a light spot would be produced on the surface of the Moon a distance of 376 × 103 Ion away from such a device? Neglect any effects of the Earth's atmosphere. Get solution

41. If you peered through a 0.75-mm hole at an eye chart, you would probably notice a decrease in visual acuity. Compute the angular limit of resolution, assuming that it's determined only by diffraction; take λ0 = 550 nm. Compare your results with the value of 1.7 × 10-4 rad, which corresponds to a 4.0-mm pupil. Get solution

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45. The neoimpressionist painter Georges Seurat was a member of the pointilhst school. His paintings consist of an enormous number of closely spaced small dots (...) of pure pigment. The illusion of color mixing is produced only in the eye of the observer. How far from such a painting should one stand in order to achieve the desired blending of color? Get solution

46. The Mount Palomar telescope has an objective mirror with a 508-em diameter. Determine its angular' limit of resolution at a wavelength of 550 nm, in radians, degrees, and seconds of arc. How far apart must two objects be on the surface of the Moon if they are to be resolvable by the Palomar telescope? The Earth-Moon distance is 3.844 × 3 08 m; take λ0 = 550 nm. How far apart must two objects be on the Moon if they are to be distinguished by the eye? Assume a pupil diameter of 4.00 mm. Get solution

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51. A transmission grating whose lines are separated by 3.0 × 10-6 m is illuininated by a narrow beam of red light (λ0 = 694.3 nm) from a ruby laser. Spots of diffracted light, on both sides of the undefeated beam, appear on a screen 2.0 m away. How far from the central axis is each of the two nearest spots? Get solution

52. A diffraction grating with slits 0.60 × 103 cm apart is illuminated by light with a wavelength of 500 nm. At what angle will the third-order maximum appear? Get solution

53. A diffraction grating produces a second-order spectrum of yellow light (λ0 = 550 nm) at 25°. Determine the spacing between the lines on the grating. Get solution

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55. White light falls normally on a transmission grating that contains 1000 lines per centimeter. At what angle will red light (λ0 = 650 nm) emerge in the first-order spectrum? Get solution

56. Light from a laboratory sodium lamp has two strong yellow (components at 589.5923 nm and 588.9953 nm. How far apart in the first-order spectrum will these two lines be on a screen 1.00 m from a grating having 10 000 lines per centimeter? Get solution

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59. Sunlight impinges on a transmission grating that is formed with 5000 lines per centimeter. Does the third-order spectrum overlap the second-order spectrum? Take red to be 780 nm and violet to be 390 nm. Get solution

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61. Light having a frequency of 4.0 × 1014 Hz is incident on a grating formed with 10 000 lines per centimeter. What is the highest-order spectrum that can be seen with this device? Explain. Get solution

62. Suppose that a grating spectrometer while in vacuum on Earth sends 500-nm light off at an angle of 20.0° in the first-order spectrum. By comparison, after landing on the planet Mongo, the same light is diffracted through 18.0°. Determine the index of refraction of the Mongoian atmosphere. Get solution

63. Prove that the equationa(sin θm - sin θ1) = mλ [10.61]when applied to a transmission grating, is independent of the refractive index. Get solution

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65. A high-resolution grating 260 nun wide, with 300 lines per millimeter, at about 75° in autocollimation has a resolving power of just about 106 for λ = 500 nm. Find its free spectral range. How do these values of R and (∆λ)fsr compare with those of a Fabry-Perot etalon having a 1-cm air gap and a finesse of 25? Get solution

66. What is the total number of lines a grating must have in order just to separate the sodium doublet (λ1 = 5895.9 Å, λ2 = 5890.0 A) in the third order? Get solution

67. Imagine an opaque screen containing 30 randomly located circular holes. The light source is such that every aperture is coherently illuminated by its own plane wave. Each wave in turn is completely incoherent with respect to all the others. Describe the resulting far-field diffraction pattern. Get solution

68. Imagine that you are looking through a piece of square woven cloth at a point source (λ0 = 600 nm) 20m away. If you see a square arrangement of bright spots located about the point source (Fig. P. 10.41), each separated by an apparent nearest-neighbor distance of 12 cm, how close together are the strands of cloth?Figure P.10.41 (Photo by E. H.)... Get solution

69. Perform the necessary mathematical operations needed to arrive at Eq. (10.76). Get solution

70. Referring to Fig. 10.38, integrate the expression dS = 2πρ2 sin (φ dφ over the lth zone to get the area of that zone,...Show that the mean distance to the lth zone is...so that the ratio Al/rl is constant Get solution

71. 'Derive Eq. (10.84). Get solution

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73. Collimated light from a krypton ion laser at 568.19 nm impinges normally on a circular aperture. When viewed axially from a distance of 1.00 m, the hole uncovers the first half-period Fresnel zone. Determine its diameter Get solution

74. Plane waves impinge perpendicularly on a screen with a small circular hole in it. It is found that when viewed from some axial point P the hole uncovers P of the first half-period zone. What is the irradiance at P in terms of the irradiance there when the screen is removed? Get solution

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83. A collimated beam from a ruby laser (694.3 nm) having an irradiance of 10 W/m2 is incident perpendicularly on an opaque screen containing a square hole 5.0 mm on a side. Compute the irradiance at a point on the central axis 250 cm from the aperture. Get solution

84. Use the Comu spiral to make a rough sketch of ... (w1+w2)/2 for ∆w = 5.5. Compare your results with those of Fig. 10.57. Get solution

85. The Fresnel integrals have the asymptotic forms (corresponding to large values of w) given by......Using this fact, show that the irradiance in the shadow of a semi-infinite opaque screen decreases in proportion to the inverse square of the distance to the edge, as z1 and therefore v1 become large. Get solution

86. What would you expect to see on the plane of observation if the half-plane 2 in Fig. 10.58 were semi-transparent? Get solution

87. Plane waves from a collimated He-Ne laserbeam (λ0 = 632.8 nm) impinge on a steel rod with a 2.5-mm diameter. Draw a rough graphic representation of the diffraction pattern that would be seen on a screen 3.16 m from the rod. Get solution

88. Make a rough sketch of the irradiance function for a Fresnel diffraction pattern arising from a double slit. What would the Cornu spiral picture look like at point P0? Get solution

89. Make a rough sketch of a possible Fresnel diffraction pattern arising from each of the indicated apertures (Fig. P. 10.50).Figure P.10.50... Get solution

90. Suppose the slit in Fig. 10.54 is made very wide. What will the Fresnel diffraction pattern look like? Get solution

91. A long narrow slit 0.10 mm wide is illuminated by light of wavelength 500 nm coming from a point source 0.90 m away. Determine me irradiance at a point 2.0 m beyond the screen when the slit is centered on, and perpendicular to, the line from the source to the point of observation. Write your answer in terms of the unobstructed irradiance. Get solution

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Chapter #13 Solutions - Optics - Eugene Hecht - 5th Edition

1. After a while, a cube of rough steel (10 cm on a side) reaches equilibrium inside a furnace al a temperature of 400°C. Knowing that...