Chapter #8 Solutions - Optics - Eugene Hecht - 5th Edition

1. Get solution

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4. Describe completely the state of polarization of each of the following waves:(a) ... (b) ...(c) ...(d) ... Get solution

5. Consider the disturbance given by the expression ... (z, t) = ... E0 sin kz. What kind of wave is it? Draw a rough sketch showing its main features. Get solution

6. Analytically, show that the superposition of an... and an...instate having different amplitudes will yield an ...state, as shown in Fig. 8.8. What must e be to duplicate that figure? Get solution

7. Write an expression for a ...state lightwave of angular frequency ω and amplitude E0 propagating along the .r-axis with its plane of vibration at an angle of 25° to the xy-plane. The disturbance is zero at t = 0 and x = 0. Get solution

8. Write an expression for a ...state lightwave of angular frequency ω and amplitude E0 propagating along a line in the xy-plane at 45° to the x-axis and having its plane of vibration corresponding to the .xy-plane. At t = 0, y = 0, and x = 0 the field is zero. Get solution

9. Write an expression for an ... state lightwave of frequency co propagating in the positive x-direction such that at t = 0 and x = 0 the ...-field points in the negative z-direction. Get solution

10. A beam of linearly polarized light with its electric field vertical impinges perpendicularly on an ideal linear polarizer with a vertical transmission axis. If the incoming beam has an irradiance of 200-W/m2, what is the irradiance of the transmitted beam? Get solution

11. Given that 300 W/m2 of light from an ordinary tungsten bulb arrives at an ideal linear polarizer. What is its radiant flux density on emerging? Get solution

12. A beam of vertically polarized linear light is perpendicularly incident on an ideal linear polarizer. Show that if its transmission axis makes an angle of 60° with the vertical only 25% of the irradiance will be transmitted by the polarizer. Get solution

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15. If light that is initially natural and of flux density Ii; passes through two sheets of HN-32 whose transmission axes are parallel, what will be the flux density of the emerging beam? Get solution

16. What will be the irradiance of the emerging beam if the analyzer of the previous problem is rotated 30°? Get solution

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18. The irradiance of a beam of natural light is 400 W/m2. It impinges on the first of two consecutive ideal linear polarizers whose transmission axes are 40.0° apart. How much light emerges from the two? Get solution

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22. As we saw in Section 8.10, substances such as sugar and insulin are optically active; they rotate the plane of polarization in proportion to both the path length and the concentration of the solution. A glass vessel is placed between a pair of crossed HN-50 linear polarizers, and 50% of the natural light incident on the first polarizer is transmitted through the second polarizer. By how much did the sugar solution in the cell rotate the light passed by the first polarizer? Get solution

23. The light from an ordinary flashlight is passed through a linear polarizer with its transmission axis vertical. The resulting beam, having an irradiance of 200 W/m2, is incident normally on a vertical HN-50 linear polarizer whose transmission axis is tilted al 30° above the horizontal. How much light is transmitted? Get solution

24. Linearly polarized light (with an irradiance of 200 W/m2) aligned with its electric-field vector at +55˚ from the vertical impinges perpendicularly on an ideal sheet polarizer whose transmission axis is at +10° from the vertical. What fraction of the incoming light emerges? Get solution

25. Two ideal linear sheet polarizers are arranged with respect to the vertical with their transmission axis at 10° and 60°, respectively. If a linearly polarized beam of light with its electric field at 40° enters the first polarizer, what fraction of its irradiance will emerge? Get solution

26. Imagine a pair of crossed polarizers with transmission axes vertical and horizontal. The beam emerging from the first polarizer has flux density I1, and of course no light passes through the analyzer (i.e., I2 = 0). Now insert a perfect linear polarizer (HN-50) with its transmission axis at 45° to the vertical between the two elements— compute I2- Think about the motion of the electrons that arc radiating in each polarizer. Get solution

27. Imagine that you have two identical perfect linear polarizers and a source of natural light. Place them one behind the other and position their transmission axes at 0° and 50°, respectively. Now insert between (hem a third linear polarizer with its transmission axes at 25°. If 1000 W/m2 of light is incident, how much will emerge with and without the middle polarizer in place? Get solution

28. Given that 200 W/m2 of randomly polarized light is incident normally on a stack of ideal linear polarizers that are positioned one behind the other with the transmission axis of the first vertical, the second at 30", the third at 60", and the fourth at 90°. How much light emerges? Get solution

29. Two HN-50 linear polarizers are positioned one behind the other. What angle should their transmission axes make if an incident unpolarized 100-W/m2 beam is to be reduced to 30.0 W/m2 on emerging from the pan'? Get solution

30. An ideal polarizer is rotated at a rate co between a similar pair of stationary crossed polarizers. Show that the emergent flux density will be modulated at four tunes the rotational frequency. In other words, show that...where I1, is the flux density emerging from the first polarizer and I is the final flux density. Get solution

31. Figure P.8.22 shows a ray traversing a calcite crystal at nearly normal incidence, bouncing off a mirror, and then going through the crystal again. Will the observer see a double image of the spot on ...?... Get solution

32. A pencil mark on a sheet of paper is covered by a calcite crystal. With illumination from above, isn't the light impinging on the paper already polarized, having passed through the crystal? Why then do we see two images? Test your solution by polarizing the light from a flashlight and then reflecting it off a sheet of paper. Try specular reflection off glass; is the reflected light polarized? Get solution

33. Discuss in detail what you see in Fig. P.8.24. The crystal in the photograph is calcite, and it has a blunt corner at the upper left. The two Polaroids have their transmission axes parallel to their short edges.... Get solution

34. The calcite crystal in Fig. P.8.25 is shown in three different orientations. Its blunt comer is on the left in (a), the lower left in (b), and the bottom in (c). The Polaroid's transmission axis is horizontal. Explain each photograph, particularly (b). Get solution

35. in discussing calcite, we pointed out that its large birefringence arises from the fact that the carbonate groups lie in parallel planesa...b...c...(normal to the optic axis). Show in a sketch and explain why the polarization of the group will be less when ... is perpendicular to the C03 plane than when ... is parallel to it. What does this mean with respect to v± and vn, that is, the wave's speeds when ... is linearly polarized perpendicular or parallel to the optic axis? Get solution

36. A beam of light enters a calcite prism from the left, as shown in Fig. P.8.36. There are three possible orientations of the optic axis of particular interest, and these correspond to the X-, y-, and z-directions. Imagine that we have three such prisms. In each case sketch the entering and emerging beams, showing the state of polarization. How can any one of these be used to determine n0 and ne1... Get solution

37. Compute the critical angle for the ordinary ray, that is, the angle for total internal reflection at the calcite-balsam layer of a Nicol prism. Get solution

38. Draw a quartz Wollaston prism, showing all pertinent rays and their polarization states. Get solution

39. Get solution

40. The prism shown in Fig. P.8.40 is known as a Rochon polarizer. Sketch all the pertinent rays, assuming(a) that it is made of calcite.(b) that it is made of quartz.(c) Why might such a device be more useful than a dichroic polarizer when functioning with high-flux density laser light?(d) What valuable feature of the Rochon is lacking in the Wollaston polarizer? Get solution

41. Imagine that we have a transmitter of microwaves that radiates a linearly polarized wave whose ...-field is known to be parallel to the dipole direction. We wish to reflect as much energy as possible off the surface of a pond (having an index of refraction of 9.0). Find the necessary incident angle and comment on the orientation of the beam. Get solution

42. At what angle will the reflection of the sky coming off the surface of a pond (n = 1.33) completely vanish when seen through a Polaroid filter? Get solution

43. What is Brewster's angle for reflection of light from the surface of a piece of glass (ng = 1.65) immersed in water (nw = 1.33)? Get solution

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45. A beam of light is reflected off the surface of some unknown liquid, and the light is examined with a linear sheet polarizer. It is found that when the central axis of the polarizer (that is, the perpendicular to the plane of the sheet) is tilted down from the vertical at an angle of 54.30°, the reflected light is completely passed, provided the transmission axis is parallel to the plane of the interface. From this information, compute the index of refraction of the liquid. Get solution

46. Light reflected from a glass (ng = 1.65) plate immersed in ethyl alcohol (ne = 1.36) is found to be completely linearly polarized. At what angle will the partially polarized beam be transmitted into the plate? Get solution

47. A beam of natural light is incident on an air-glass interface (nti = 1.5) at 40˚. Compute the degree of polarization of the reflected light. Get solution

48. Get solution

49. A beam of natural light incident in air on a glass (n = 1.5) interface at 70° is partially reflected. Compute the overall reflectance. How would this compare with the case of incidence at, say, 56.3°? Explain. Get solution

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52. A ray of yellow light is incident on a calcite plate at 50°. The plate is cut so that the optic axis is parallel to the front face and perpendicular to the plane-of-incidence. Find the angular separation between the two emerging rays. Get solution

53. A beam of light is incident normally on a quartz plate whose optic axis is perpendicular to the beam, If λ0 = 589.3 nm, compute the wavelengths of both the ordinary and extraordinary waves. What are their frequencies? Get solution

54. The electric-field vector of an incident ...state makes an angle of+30° with the horizontal fast axis of a quarter-wave plate. Describe, in detail, the state of polarization of the emergent wave. Get solution

55. Take two ideal Polaroids (the first with its axis vertical and the second, horizontal) and insert between them a stack of 10 half-wave plates, the first with its fast axis rotated π/40 rad from the vertical, and each subsequent one rotated π/40 rad from the previous one. Determine the ratio of the emerging to incident irradiance, showing your logic clearly. Get solution

56. Suppose you were given a linear polarizer and a quarter-wave plate. How could you determine which was which, assuming you also had a source of natural light? Get solution

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66. An ... state traverses an eighth-wave plate having a horizontal fast axis. What is its polarization state on emerging? Get solution

67. Figure P.8.44 shows two Polaroid linear polarizers and between them a microscope slide to which is attached a piece of cellophane tape. Explain what you see.... Get solution

68. Imagine that we have randomly polarized room light incident almost normally on the glass surface of a radar screen. A portion of it would be specularly reflected back toward the viewer and would thus tend to obscure the display. Suppose now that we cover the screen with a right-circular polarizer, as shown in Fig. P.8.46. Trace the incident and reflected beams, indicating their polarization states. What happens to the reflected beam?... Get solution

69. A Babinet compensator is positioned at 45° between crossed linear polarizers and is being illuminated with sodium light. When a thin sheet of mica (indices 1.599 and 1.594) is placed on the compensator, the black bands all shift by ... of the space separating them. Compute the retardance of the sheet and its thickness. Get solution

70. Is it possible for a beam to consist of two orthogonal incoherent ...-states and not be natural light? Explain. How might you arrange to have such a beam? Get solution

71. The specific rotatory power for sucrose dissolved in water at 20°C (λo = 589.3 nm) is +66.45° per 10 cm of path traversed through a solution containing 1 g of active substance (sugar) per em of solution. A vertical ...-state (sodium light) enters at one end of a 1-in tube containing 1000 cm3 of solution, of which 10 g is sucrose. At what orientation will the ...-state emerge? Get solution

72. On examining a piece of stressed photoelastic material between crossed linear polarizers, we would see a set of colored bands (isochromatics) and, superimposed on these, a set of dark bands (iso-clinics). How might we remove the isoclinics, leaving only the isochromatics? Explain your solution. Incidentally, the proper arrangement is independent of the orientation of the photoelastic sample. Get solution

73. Consider a Kerr cell whose plates are separated by a distance d. Let ℓ be the effective length of those plates (slightly different from the actual length because of fringing of the field). Show that... [8.41] Get solution

74. Compute the half-wave voltage for a longitudinal Pockels cell made of ADA (ammonium dihydrogen arsenate) at λ0 ≈ 550 nm, where r63 = 5.5 X 10-12 and no = 1.58 Get solution

75. The Jones vector for an arbitrary linearly polarized state at an angle Ɵ with respect to the horizontal is...Prove that this matrix is in agreement with the one in Table 8.5 for a. ...-state at +45°. Get solution

76. Find a Jones vector E2 representing a polarization state orthogonal to...Sketch both of these. Get solution

77. Two incoherent light beams represented by (1, 1,0,0) and (3, 0, 0, 3) are superimposed.(a) Describe in detail the polarization states of each of these.(b) Determine the resulting Stokes parameters of the combined beam and describe its polarization state.(c) What is its degree of polarization?(d) What is the resulting light produced by overlapping the incoherent beams (1, 1, 0, 0) and (1, -1, 0, 0)? Explain. Get solution

78. Show by direct calculation, using Mueller matrices, that a unit-irradiance beam of natural light passing through a vertical linear polarizer is converted into a vertical ...-state. Determine its relative irradiance and degree of polarization. Get solution

79. Show by direct calculation, using Mueller matrices, that a unit-irradiance beam of natural light passing through a linear polarizer with its transmission axis at +45° is converted into a ...-state al +45°. Determine its relative irradiance and degree of polarization. Get solution

80. Show by direct calculation, using Mueller matrices, that a beam of horizontal ...-state light passing through a ...λ-plate with its fast axis horizontal emerges unchanged. Get solution

81. Confirm that the matrix...will serve as a Mueller matrix for a quarter-wave plate with its fast axis at +45°. Shine linear light polarized at 45° through it. What happens? What emerges when a horizontal ...-state enters the device? Get solution

82. The Mueller matrix...in which C = cos 2α and S = sin 2 α, represents an arbitrary wave-plate having a retardance ∆φ and a fast axis at an angle α measured with respect to the horizontal. Use it to derive the matrix given in the previous problem. Get solution

83. Beginning with the Mueller matrix for an arbitrary retarder provided in the previous problem, show that it agrees with the matrix in Table 8.6 for a quarter-wave plate with a vertical fast axis. Get solution

84. Derive the Mueller matrix for a quarter-wave plate with its fast axis at -45°. Check that this matrix effectively cancels the one ir Problem 8.58, so that a beam passing through the two wave *** successively remains unaltered. Get solution

85. Pass a beam of horizontally polarized '' each one of the Δ-plates in the two previous the states of the emerging light. Explain which field component is leading which and how Fig. 8.7 compares with these results. Get solution

86. Use Table 8.6 to derive a Mueller matrix for a half-wave plate having a vertical fast axis. Utilize your result to convert an ... into an .... Verify that the same wave plate will convert an ... to an -.... Advancing or retarding the relative phase by π/2 should have the same effect. Check this by deriving the matrix for a half-wave plate with a horizontal fast axis. Get solution

87. Construct one possible Mueller matrix for a right-circular polarizer made out of a linear polarizer and a quarter-wave plate. Such a device is obviously an inhomogeneous two-element train and will differ from the homogeneous circular polarizer of Table 8.6. Test your matrix to determine that it will convert natural light to an .... Show that it will pass . ..., as will the homogeneous matrix. Your matrix should convert instates incident on the input side to . ...., whereas the homogeneous polarizer will totally absorb them. Verify this. Get solution

88. If the Pockets cell modulator shown in Fig. 8.57 is illuminated by light of irradiance Ii it will transmit a beam of irradiance I, such that...Make a plot of ItIi, versus applied voltage. What is the significance of the voltage that corresponds to maximum transmission? What is the lowest voltage above zero that will cause It to be zero for ADP (λ0 = 546.1 nm)? How can things be rear-ranged to yield a maximum value of ItIi for zero voltage? In this new configuration what irradiance results when V — Vλ/2? Get solution

89. Construct a Jones matrix for an isotropic plate of absorbing material having an amplitude transmission coefficient of t. It might sometimes be desirable to keep track of the phase, since even if t=1. such a plate is still an isotropic phase retarder. What is the Jones matrix for a region of vacuum? What is it for a perfect absorber? Get solution

90. Construct a Mueller matrix for an isotropic plate of absorbing material having an amplitude transmission coefficient of t What Mueller matrix will completely depolarize any wave without affecting its irradiance? (It has no physical counterpart.) Get solution

91. Keeping Eq. (8.29) in mind, write an expression for the randomly polarized flux density component (In) of a partially polarized beam in terms of the Stokes parameters. To check your result, add a randomly polarized Stokes vector of flux density 4 to an . ... of flux density 1. Then see if you get In = 4 for the resultant wave. Get solution

92. An optical filter can be described by a Jones matrix...Obtain the form of the emerging light for each of the following incident beams:(a) A plane polarized beam polarized at angle θ to the horizontal (see Problem 8.52).(b) A left-circularly polarized beam.(c) A right-circularly polarized beam.(d) From the above, identify the filter and explain how it could be constructed. Get solution

93. An optical Filter can be described by a Jones matrix...(a) Obtain the form of the emerging beam when the incident light is plane poloarized at angle θ to the horizontal (see Problem 8.52),(b) Deduce from the result of part (a) the nature of the filter.(c) Confirm your deduction above with at least one other test. Get solution

94. Two linear optical filters have Jones matrices...Identify these filters. Get solution

95. A liquid cell containing an optically active sugar solution has a Jones matrix given by...(a) Determine the polarization of the emerging light if the incident beam is a horizontal ....(b) Determine the polarization of the emerging light if the incident beam is a vertical ....(c) Determine the angle of rotation produced by the optically active material. Get solution


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...