20. A beam of light always spreads out. Why can a beam not be created with parallel rays to prevent spreading? Why can lenses, mirrors, or
apertures not be used to correct the spreading?
21. What effect does increasing the wedge angle have on the spacing of interference fringes? If the wedge angle is too large, fringes are not
observed. Why?
22. How is the difference in paths taken by two originally in-phase light waves related to whether they interfere constructively or destructively? How
can this be affected by reflection? By refraction?
23. Is there a phase change in the light reflected from either surface of a contact lens floating on a person’s tear layer? The index of refraction of the
lens is about 1.5, and its top surface is dry.
24. In placing a sample on a microscope slide, a glass cover is placed over a water drop on the glass slide. Light incident from above can reflect from
the top and bottom of the glass cover and from the glass slide below the water drop. At which surfaces will there be a phase change in the reflected
light?
25. Answer the above question if the fluid between the two pieces of crown glass is carbon disulfide.
26. While contemplating the food value of a slice of ham, you notice a rainbow of color reflected from its moist surface. Explain its origin.
27. An inventor notices that a soap bubble is dark at its thinnest and realizes that destructive interference is taking place for all wavelengths. How
could she use this knowledge to make a non-reflective coating for lenses that is effective at all wavelengths? That is, what limits would there be on
the index of refraction and thickness of the coating? How might this be impractical?
28. A non-reflective coating like the one described in Example 27.6 works ideally for a single wavelength and for perpendicular incidence. What happens for other wavelengths and other incident directions? Be specific.
29. Why is it much more difficult to see interference fringes for light reflected from a thick piece of glass than from a thin film? Would it be easier if
monochromatic light were used?
30. Under what circumstances is the phase of light changed by reflection? Is the phase related to polarization?
31. Can a sound wave in air be polarized? Explain.
32. No light passes through two perfect polarizing filters with perpendicular axes. However, if a third polarizing filter is placed between the original
two, some light can pass. Why is this? Under what circumstances does most of the light pass?
33. Explain what happens to the energy carried by light that it is dimmed by passing it through two crossed polarizing filters.
34. When particles scattering light are much smaller than its wavelength, the amount of scattering is proportional to 1 / λ 4 . Does this mean there is
more scattering for small λ than large λ ? How does this relate to the fact that the sky is blue?
35. Using the information given in the preceding question, explain why sunsets are red.
36. When light is reflected at Brewster’s angle from a smooth surface, it is 100% polarized parallel to the surface. Part of the light will be refracted
into the surface. Describe how you would do an experiment to determine the polarization of the refracted light. What direction would you expect the
polarization to have and would you expect it to be 100% ?
27.9 *Extended Topic* Microscopy Enhanced by the Wave Characteristics of Light
37. Explain how microscopes can use wave optics to improve contrast and why this is important.
38. A bright white light under water is collimated and directed upon a prism. What range of colors does one see emerging?
994 CHAPTER 27 | WAVE OPTICS
Problems & Exercises
27.1 The Wave Aspect of Light: Interference
1. Show that when light passes from air to water, its wavelength
decreases to 0.750 times its original value.
2. Find the range of visible wavelengths of light in crown glass.
3. What is the index of refraction of a material for which the wavelength of
light is 0.671 times its value in a vacuum? Identify the likely substance.
4. Analysis of an interference effect in a clear solid shows that the
wavelength of light in the solid is 329 nm. Knowing this light comes from
a He-Ne laser and has a wavelength of 633 nm in air, is the substance
zircon or diamond?
5. What is the ratio of thicknesses of crown glass and water that would
contain the same number of wavelengths of light?
Figure 27.56 The distance between adjacent fringes is Δ y = xλ / d , assuming the
27.3 Young’s Double Slit Experiment
slit separation d is large compared with λ .
6. At what angle is the first-order maximum for 450-nm wavelength blue
19. Using the result of the problem above, calculate the distance between
light falling on double slits separated by 0.0500 mm?
fringes for 633-nm light falling on double slits separated by 0.0800 mm,
located 3.00 m from a screen as in Figure 27.56.
7. Calculate the angle for the third-order maximum of 580-nm wavelength
yellow light falling on double slits separated by 0.100 mm.
20. Using the result of the problem two problems prior, find the
wavelength of light that produces fringes 7.50 mm apart on a screen 2.00
8. What is the separation between two slits for which 610-nm orange light
m from double slits separated by 0.120 mm (see Figure 27.56).
has its first maximum at an angle of 30.0º ?
9. Find the distance between two slits that produces the first minimum for
27.4 Multiple Slit Diffraction
410-nm violet light at an angle of 45.0º .
21. A diffraction grating has 2000 lines per centimeter. At what angle will
the first-order maximum be for 520-nm-wavelength green light?
10. Calculate the wavelength of light that has its third minimum at an
angle of 30.0º when falling on double slits separated by 3.00 μm .
22. Find the angle for the third-order maximum for 580-nm-wavelength
yellow light falling on a diffraction grating having 1500 lines per
Explicitly, show how you follow the steps in Problem-Solving Strategies
centimeter.
23. How many lines per centimeter are there on a diffraction grating that
11. What is the wavelength of light falling on double slits separated by
2.00 μm
gives a first-order maximum for 470-nm blue light at an angle of 25.0º ?
if the third-order maximum is at an angle of 60.0º ?
24. What is the distance between lines on a diffraction grating that
12. At what angle is the fourth-order maximum for the situation in the first
produces a second-order maximum for 760-nm red light at an angle of
problem?
60.0º ?
13. What is the highest-order maximum for 400-nm light falling on double
25. Calculate the wavelength of light that has its second-order maximum
slits separated by 25.0 μm ?
at 45.0º when falling on a diffraction grating that has 5000 lines per
14. Find the largest wavelength of light falling on double slits separated
centimeter.
by 1.20 μm for which there is a first-order maximum. Is this in the
26. An electric current through hydrogen gas produces several distinct
visible part of the spectrum?
wavelengths of visible light. What are the wavelengths of the hydrogen
spectrum, if they form first-order maxima at angles of 24.2º , 25.7º ,
15. What is the smallest separation between two slits that will produce a
second-order maximum for 720-nm red light?
29.1º , and 41.0º when projected on a diffraction grating having 10,000
lines per centimeter? Explicitly show how you follow the steps in
16. (a) What is the smallest separation between two slits that will produce Problem-Solving Strategies for Wave Optics
a second-order maximum for any visible light? (b) For all visible light?
27. (a) What do the four angles in the above problem become if a
17. (a) If the first-order maximum for pure-wavelength light falling on a
5000-line-per-centimeter diffraction grating is used? (b) Using this
double slit is at an angle of 10.0º , at what angle is the second-order
grating, what would the angles be for the second-order maxima? (c)
maximum? (b) What is the angle of the first minimum? (c) What is the
Discuss the relationship between integral reductions in lines per
highest-order maximum possible here?
centimeter and the new angles of various order maxima.
18. Figure 27.56 shows a double slit located a distance x from a
28. What is the maximum number of lines per centimeter a diffraction
screen, with the distance from the center of the screen given by y .
grating can have and produce a complete first-order spectrum for visible
light?
When the distance d between the slits is relatively large, there will be
numerous bright spots, called fringes. Show that, for small angles (where
29. The yellow light from a sodium vapor lamp seems to be of pure
sin θ ≈ θ , with θ in radians), the distance between fringes is given by wavelength, but it produces two first-order maxima at 36.093º and
Δ y = xλ / d .
36.129º when projected on a 10,000 line per centimeter diffraction
grating. What are the two wavelengths to an accuracy of 0.1 nm?
30. What is the spacing between structures in a feather that acts as a
reflection grating, given that they produce a first-order maximum for
525-nm light at a 30.0º angle?
31. Structures on a bird feather act like a reflection grating having 8000
lines per centimeter. What is the angle of the first-order maximum for
600-nm light?
CHAPTER 27 | WAVE OPTICS 995
32. An opal such as that shown in Figure 27.17 acts like a reflection
from the grating to the screen or detector. Discuss the practicality of the
grating with rows separated by about 8 μm . If the opal is illuminated
device in terms of being able to discern between wavelengths of interest.
normally, (a) at what angle will red light be seen and (b) at what angle will
blue light be seen?
33. At what angle does a diffraction grating produces a second-order
43. (a) At what angle is the first minimum for 550-nm light falling on a
maximum for light having a first-order maximum at 20.0º ?
single slit of width 1.00 μm ? (b) Will there be a second minimum?
34. Show that a diffraction grating cannot produce a second-order
44. (a) Calculate the angle at which a 2.00-μm -wide slit produces its
maximum for a given wavelength of light unless the first-order maximum
is at an angle less than 30.0º .
first minimum for 410-nm violet light. (b) Where is the first minimum for
700-nm red light?
35. If a diffraction grating produces a first-order maximum for the shortest
45. (a) How wide is a single slit that produces its first minimum for
wavelength of visible light at 30.0º , at what angle will the first-order
633-nm light at an angle of 28.0º ? (b) At what angle will the second
maximum be for the longest wavelength of visible light?
minimum be?
36. (a) Find the maximum number of lines per centimeter a diffraction
46. (a) What is the width of a single slit that produces its first minimum at
grating can have and produce a maximum for the smallest wavelength of
60.0º for 600-nm light? (b) Find the wavelength of light that has its first
visible light. (b) Would such a grating be useful for ultraviolet spectra? (c)
For infrared spectra?
minimum at 62.0º .
37. (a) Show that a 30,000-line-per-centimeter grating will not produce a
47. Find the wavelength of light that has its third minimum at an angle of
maximum for visible light. (b) What is the longest wavelength for which it
48.6º when it falls on a single slit of width 3.00 μm .
does produce a first-order maximum? (c) What is the greatest number of
lines per centimeter a diffraction grating can have and produce a
48. Calculate the wavelength of light that produces its first minimum at an
complete second-order spectrum for visible light?
angle of 36.9º when falling on a single slit of width 1.00 μm .
38. A He–Ne laser beam is reflected from the surface of a CD onto a
49. (a) Sodium vapor light averaging 589 nm in wavelength falls on a
wall. The brightest spot is the reflected beam at an angle equal to the
angle of incidence. However, fringes are also observed. If the wall is 1.50
single slit of width 7.50 μm . At what angle does it produces its second
m from the CD, and the first fringe is 0.600 m from the central maximum,
minimum? (b) What is the highest-order minimum produced?
what is the spacing of grooves on the CD?
50. (a) Find the angle of the third diffraction minimum for 633-nm light
39. The analysis shown in the figure below also applies to diffraction
falling on a slit of width 20.0 μm . (b) What slit width would place this
gratings with lines separated by a distance d . What is the distance
minimum at 85.0º ? Explicitly show how you follow the steps in
between fringes produced by a diffraction grating having 125 lines per
centimeter for 600-nm light, if the screen is 1.50 m away?
Problem-Solving Strategies for Wave Optics
51. (a) Find the angle between the first minima for the two sodium vapor
lines, which have wavelengths of 589.1 and 589.6 nm, when they fall
upon a single slit of width 2.00 μm . (b) What is the distance between
these minima if the diffraction pattern falls on a screen 1.00 m from the
slit? (c) Discuss the ease or difficulty of measuring such a distance.
52. (a) What is the minimum width of a single slit (in multiples of λ ) that
will produce a first minimum for a wavelength λ ? (b) What is its
minimum width if it produces 50 minima? (c) 1000 minima?
53. (a) If a single slit produces a first minimum at 14.5º , at what angle is
the second-order minimum? (b) What is the angle of the third-order
minimum? (c) Is there a fourth-order minimum? (d) Use your answers to
illustrate how the angular width of the central maximum is about twice the
angular width of the next maximum (which is the angle between the first
and second minima).
Figure 27.57 The distance between adjacent fringes is Δ y = xλ / d , assuming the 54. A double slit produces a diffraction pattern that is a combination of single and double slit interference. Find the ratio of the width of the slits
slit separation d is large compared with λ .
to the separation between them, if the first minimum of the single slit
40. Unreasonable Results
pattern falls on the fifth maximum of the double slit pattern. (This will
greatly reduce the intensity of the fifth maximum.)
Red light of wavelength of 700 nm falls on a double slit separated by 400
nm. (a) At what angle is the first-order maximum in the diffraction
55. Integrated Concepts
pattern? (b) What is unreasonable about this result? (c) Which
A water break at the entrance to a harbor consists of a rock barrier with a
assumptions are unreasonable or inconsistent?
50.0-m-wide opening. Ocean waves of 20.0-m wavelength approach the
41. Unreasonable Results
opening straight on. At what angle to the incident direction are the boats
inside the harbor most protected against wave action?
(a) What visible wavelength has its fourth-order maximum at an angle of
25.0º when projected on a 25,000-line-per-centimeter diffraction
56. Integrated Concepts
grating? (b) What is unreasonable about this result? (c) Which
An aircraft maintenance technician walks past a tall hangar door that acts
assumptions are unreasonable or inconsistent?
like a single slit for sound entering the hangar. Outside the door, on a line
perpendicular to the opening in the door, a jet engine makes a 600-Hz
42. Construct Your Own Problem
sound. At what angle with the door will the technician observe the first
Consider a spectrometer based on a diffraction grating. Construct a
minimum in sound intensity if the vertical opening is 0.800 m wide and
problem in which you calculate the distance between two wavelengths of
the speed of sound is 340 m/s?
electromagnetic radiation in your spectrometer. Among the things to be
considered are the wavelengths you wish to be able to distinguish, the
27.6 Limits of Resolution: The Rayleigh Criterion
number of lines per meter on the diffraction grating, and the distance
996 CHAPTER 27 | WAVE OPTICS
57. The 300-m-diameter Arecibo radio telescope pictured in Figure 27.28 of 35 cm; find the minimum separation of two dots such that they cannot detects radio waves with a 4.00 cm average wavelength.
be resolved. How many dots per inch (dpi) does this correspond to?
(a) What is the angle between two just-resolvable point sources for this
68. Unreasonable Results
telescope?
An amateur astronomer wants to build a telescope with a diffraction limit
(b) How close together could these point sources be at the 2 million light
that will allow him to see if there are people on the moons of Jupiter.
year distance of the Andromeda galaxy?
(a) What diameter mirror is needed to be able to see 1.00 m detail on a
58. Assuming the angular resolution found for the Hubble Telescope in
Jovian Moon at a distance of 7.50×108 km from Earth? The
Example 27.5, what is the smallest detail that could be observed on the
Moon?
wavelength of light averages 600 nm.
59. Diffraction spreading for a flashlight is insignificant compared with
(b) What is unreasonable about this result?
other limitations in its optics, such as spherical aberrations in its mirror.
(c) Which assumptions are unreasonable or inconsistent?
To show this, calculate the minimum angular spreading of a flashlight
69. Construct Your Own Problem
beam that is originally 5.00 cm in diameter with an average wavelength of
600 nm.
Consider diffraction limits for an electromagnetic wave interacting with a
circular object. Construct a problem in which you calculate the limit of
60. (a) What is the minimum angular spread of a 633-nm wavelength He-
angular resolution with a device, using this circular object (such as a lens,
Ne laser beam that is originally 1.00 mm in diameter?
mirror, or antenna) to make observations. Also calculate the limit to
(b) If this laser is aimed at a mountain cliff 15.0 km away, how big will the
spatial resolution (such as the size of features observable on the Moon)
illuminated spot be?
for observations at a specific distance from the device. Among the things
(c) How big a spot would be illuminated on the Moon, neglecting
to be considered are the wavelength of electromagnetic radiation used,
atmospheric effects? (This might be done to hit a corner reflector to
the size of the circular object, and the distance to the system or
measure the round-trip time and, hence, distance.) Explicitly show how
phenomenon being observed.
you follow the steps in Problem-Solving Strategies for Wave Optics.
61. A telescope can be used to enlarge the diameter of a laser beam and
limit diffraction spreading. The laser beam is sent through the telescope
70. A soap bubble is 100 nm thick and illuminated by white light incident
in opposite the normal direction and can then be projected onto a satellite perpendicular to its surface. What wavelength and color of visible light is
or the Moon.
most constructively reflected, assuming the same index of refraction as
(a) If this is done with the Mount Wilson telescope, producing a 2.54-m-
water?
diameter beam of 633-nm light, what is the minimum angular spread of
71. An oil slick on water is 120 nm thick and illuminated by white light
the beam?
incident perpendicular to its surface. What color does the oil appear
(b) Neglecting atmospheric effects, what is the size of the spot this beam
(what is the most constructively reflected wavelength), given its index of
refraction is 1.40?
would make on the Moon, assuming a lunar distance of 3.84×108 m ?
72. Calculate the minimum thickness of an oil slick on water that appears
62. The limit to the eye’s acuity is actually related to diffraction by the
blue when illuminated by white light perpendicular to its surface. Take the
pupil.
blue wavelength to be 470 nm and the index of refraction of oil to be
(a) What is the angle between two just-resolvable points of light for a
1.40.
3.00-mm-diameter pupil, assuming an average wavelength of 550 nm?
73. Find the minimum thickness of a soap bubble that appears red when
(b) Take your result to be the practical limit for the eye. What is the
illuminated by white light perpendicular to its surface. Take the
greatest possible distance a car can be from you if you can resolve its
wavelength to be 680 nm, and assume the same index of refraction as
two headlights, given they are 1.30 m apart?
water.
(c) What is the distance between two just-resolvable points held at an
74. A film of soapy water ( n = 1.33 ) on top of a plastic cutting board
arm’s length (0.800 m) from your eye?
has a thickness of 233 nm. What color is most strongly reflected if it is
(d) How does your answer to (c) compare to details you normally observe illuminated perpendicular to its surface?
i