P7.34.f^Chapter 34 Ending^1122^1131^,,^46517^47033%
WHAT WE HAVE LEARNED
EXAM STUDY GUIDE
KEY TERMS
Huygens's Principle, p. 1098 coherent light, p. 1100 incoherent, p. 1100 interference, p. 1100 constructive interference, p. 1100 destructive interference, p. 1100 Young's double-slit experiment, p. 1101 order, p. 1102 thin film, p. 1104 Newton's rings, p. 1106 interferometer, p. 1107 diffraction, p. 1109 resolution, p. 1113 Rayleigh's Criterion, p. 1113 diffraction grating, p. 1115 dispersion, p. 1116 resolving power, p. 1117 X-ray diffraction, p. 1121 Bragg's Law, p. 1121
NEW SYMBOLS AND EQUATIONS

Δx = mλ (m = 0,±1,±2, …), path length difference for constructive interference

(m = 0,± 1,± 2,…) path length difference for destructive interference

N, number of rulings in a diffraction grating

nl, number of rulings per unit length for a diffraction grating

(m = 1,2,3,…), dispersion of a diffraction grating

R = Nm (m = 1,2,3,…), resolving power of a diffraction grating

ANSWERS TO SELF-TEST OPPORTUNITES
  • 34.1

    There is no change in colors. After all, the light has to propagate through your eyeball before it reaches your retina, and the index of refraction of your eyeball does not change when your head is underwater.

  • 34.2
  • 34.3

    Assume that λ = 550 nm and that the diameter of the human pupil is d = 10.0 mm.

  • Individual watchtowers on the Great Wall of China are difficult to see from the orbiting Space Shuttle.

  • 34.4

    We get for the first maximum θ1 = sin−1 (532 nm/740 nm) = 46.0°. No other maxima are possible.

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PROBLEM SOLVING PRACTICE
Problem-Solving Guidelines
  1. A sketch of the optical situation is almost always helpful. It is simplest to use rays, not wave fronts, but remember that you are dealing with wave effects. Use the diagram to clearly identify any path differences involved in the problem.

  2. The basic idea of wave optics is that constructive interference occurs when the path difference is an integer number of wavelengths, while destructive interference occurs when the path difference is an odd-integer number of half wavelengths. Always start from this concept and take into account in any additional phase changes due to reflection.

  3. Remember that a phase change occurs when light reflects from a more dense medium after being incident in a less dense medium. If light reflects from a less dense medium, no phase change occurs.

SOLVED PROBLEM34.2 Spy Satellite

You have been assigned the job of designing a camera lens for a spy satellite. This satellite will orbit the Earth at an altitude of 201 km. The camera is sensitive to light with a wavelength of 607 nm. The camera must be able to resolve objects on the ground that are 0.490 m apart.

PROBLEM

What is the minimum diameter of the lens?

SOLUTION
THINK

This camera lens will be limited by diffraction. We can apply Rayleigh's Criterion to calculate the minimum diameter of the lens, given the angle subtended by two objects on the ground as viewed from the spy satellite orbiting above.

SKETCH
FIGURE 34.53
A spy satellite at height h observing two objects on the ground a distance Δx apart.

Figure 34.53 shows a sketch of the spy satellite observing two objects on the ground

RESEARCH

The Rayleigh Criterion for resolving two objects separated by an angle θR using light with wavelength λ is given by

where d is the diameter of the circular camera lens in the spy satellite.

The angle required for the performance requirement of the spy satellite is given by

where Δx is the distance between the two objects on the ground and h is the height of the spy satellite above the ground.

SIMPLIFY

We can equate θR and θs to get

Solving for the diameter of the camera lens gives us

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CALCULATE

Putting in the numerical values gives us

ROUND

We report our result to three significant figures,

DOUBLE-CHECK

To double-check our result, we make the small-angle approximation for the Rayleigh Criterion and the angle subtended by the two objects. For the case where the wavelength of light is much smaller than the aperture of the camera, we can write

For the angle seen by the camera in the spy satellite, we can write

Thus,

which can be solved for the minimum diameter of the camera lens

which agrees with our result within round-off errors. Thus, our result seems reasonable.

SOLVED PROBLEM34.3 Air Wedge

Light of wavelength λ = 516 nm is incident perpendicularly on two glass plates. The glass plates are spaced at one end by a thin piece of kapton film. Due to the wedge of air created by this film, 25 bright interference fringes are observed across the top plate, with a dark fringe at the end by the film.

PROBLEM

How thick is the film?

SOLUTION
THINK

Light passes through the top plate, reflects from the top surface of the bottom plate, and then interferes with light reflected from the bottom surface of the top plate. A phase change occurs when the light is reflected from the bottom plate, so the criterion for constructive interference is that the path length is equal to an integer plus one-half times the wavelength. The criterion for destructive interference is that the path difference is an integer times the wavelength.

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SKETCH
FIGURE 34.54
Two glass plates with a thin film separating the plates at one end. The green arrows represent the incident light from the top. The angle of the light is exaggerated for clarity. The interfering light is reflected off the bottom of the top plate and the top of the bottom plate.

Figure 34.54 is a sketch showing the two glass plates with a thin piece of film separating the plates at one end. Light is incident vertically from the top.

RESEARCH

At any point along the plates, the criterion for constructive interference is given by

where t is the separation between the plates, m is an integer, and λ is the wavelength of the incident light. There are 25 bright fringes visible. The first bright fringe corresponds to m = 0, and the 25th bright fringe corresponds to m = 24.

Immediately past the 25th is a dark fringe where the piece of film is located. The criterion for destructive interference is 2t = nλ, where n is an integer.

SIMPLIFY

The dark fringe located at the end of the glass plate where the film is located is given by

where the factor describes the constructive interference and the extra produces the dark fringe at the end of the plate with the film. Thus we can solve for the separation of the plates, which corresponds to the thickness of the film:

CALCULATE

Putting in the numerical values gives

ROUND

We report our result to three significant figures,

DOUBLE-CHECK

Occasionally throughout this book we need to add reminders that checking our results just for the right units and expected order of magnitude can do a lot to prevent simple errors. Here the unit m is certainly the right one for a physical length, in this case the film thickness. At first glance you may think that ~10−5 m, on the order of 1/100th of the thickness of a fingernail, may be impossibly thin for a solid film. However, an Internet search on kapton film will show that 6.5 μm is well within the range of thicknesses in which kapton film is produced. Thus, our answer seems plausible.

MULTIPLE-CHOICE QUESTIONS
QUESTIONS
Page 1128
PROBLEMS

A blue problem number indicates a worked-out solution is available in the Student Solutions Manual. One and two •• indicate increasing level of problem difficulty.

Sections 34.1
  • 34.19 A helium-neon laser has a wavelength of 632.8 nm.

    1. What is the wavelength of this light as it passes through Lucite with an index of refraction n = 1.500?

    2. What is the speed of light in the Lucite?

    Answer

  • 34.20 It is common knowledge that the visible light spectrum extends approximately from 400 nm to 700 nm. Roughly, 400 nm to 500 nm corresponds to blue light, 500 nm to 550 nm corresponds to green, 550 nm to 600 nm to yellow-orange, and above 600 nm to red. In an experiment, red light with a wavelength of 632.8 nm from a HeNe laser is refracted into a fish tank filled with water with index of refraction 1.333. What is the wavelength of the same laser beam in water, and what color will the laser beam have in water?

Sections 34.2 and 34.3
  • 34.21 What minimum path difference is needed to cause a phase shift by π/4 in light of wavelength 700. nm?

    Answer

  • 34.22 Coherent, monochromatic light of wavelength 450.0 nm is emitted from two locations and detected at another location. The path difference between the two routes taken by the light is 20.25 cm. Will the two light waves interfere destructively or constructively at the detection point?

  • 34.23 A Young's interference experiment is performed with monochromatic green light (λ = 540 nm). The separation between the slits is 0.100 mm, and the interference pattern on a screen shows the first side maximum 5.40 mm from the center of the pattern. How far away from the slits is the screen?

    Answer

  • 34.24 For a double-slit experiment, two 1.50-mm wide slits are separated by a distance of 1.00 mm. The slits are illuminated by a laser beam with wavelength 633 nm. If a screen is placed 5.00 m away from the slits, determine the separation of the bright fringes on the screen.

  • 34.25 Coherent monochromatic light with wavelength λ = 514 nm is incident on two slits that are separated by a distance d = 0.500 mm. The intensity of the radiation at a screen 2.50 m away from each slit is 180.0 W/cm2. Determine the position y1/3 at which the intensity of the central peak (at y = 0) drops to Imax/3.

    Answer

  • 34.26 In a double-slit experiment, He-Ne laser light of wavelength 633 nm produced an interference pattern on a screen placed at some distance from the slits. When one of the slits was covered with a thin glass slide of thickness 12.0 μm, the central fringe shifted to the point occupied earlier by the 10th dark fringe (see figure). What is the refractive index of the glass slide?

Sections 34.4
  • 34.27 Suppose the thickness of a thin soap film (n = 1.32) surrounded by air is nonuniform and gradually tapers. Monochromatic light of wavelength 550 nm illuminates the film. At the thinnest end, a dark band is observed. How thick is the film at the next two dark bands closest to the first dark band?

    Answer

  • 34.28 White light (400 nm <λ < 750 nm) shines onto a puddle of water (n = 1.33). There is a thin (100.0 nm thick) layer of oil (n = 1.47) on top of the water. What wavelengths of light would you see reflected?

  • Page 1129
  • 34.29 Some mirrors for infrared lasers are constructed with alternating layers of hafnia and silica. Suppose you want to produce constructive interference from a thin film of hafnia (n = 1.90) on BK-7 glass (n = 1.51) when infrared radiation of wavelength 1.06 μm is used. What is the smallest film thickness that would be appropriate, assuming the laser beam is oriented at right angles to the film?

    Answer

  • 34.30 Sometimes thin films are used as filters to prevent certain colors from entering a lens. Consider an infrared filter, designed to prevent 800.0-nm light from entering a lens. Find the minimum film thickness for a layer of MgF2 (n = 1.38) to prevent this light from entering the lens.

  • •34.31 White light shines on a sheet of mica that has a uniform thickness of 1.30 μm. When the reflected light is viewed using a spectrometer, it is noted that light with wavelengths of 433.3 nm, 487.5 nm, 557.1 nm, 650.0 nm, and 780.0 nm is not present in the reflected light. What is the index of refraction of the mica?

    Answer

  • 34.32 A single beam of coherent light (λ = 633 · 10−9 m) is incident on two glass slides, which are touching at one end and are separated by a 0.0200-mm thick sheet of paper on the other end, as shown in the figure below. Beam 1 reflects off the bottom surface of the top slide, and Beam 2 reflects off the top surface of the bottom slide. Assume that all the beams are perfectly vertical and that they are perpendicular to both slides, i.e., the slides are nearly parallel (the angle is exaggerated in the figure); the beams are shown at angles in the figure so that they are easier to identify. Beams 1 and 2 recombine at the location of the eye in the figure below. The slides are 8.00 cm long. Starting from the left end (x = 0), at what positions xbright do bright bands appear to the observer above the slides? How many bright bands are observed?

  • 34.33 A common interference setup consists of a plano-convex lens placed on a plane mirror and illuminated from above at normal incidence with monochromatic light. The pattern of circular interference fringes (fringes of equal thickness)—bright and dark circles—formed due to the air wedge defined by the two glass surfaces, is known as Newton's rings. In an experiment using a plano-convex lens with focal length f = 80.00 cm and index of refraction n1 = 1.500, the radius of the third bright circle is found to be 0.8487 mm. Determine the wavelength of the monochromatic light.

    Answer

Section 34.5
  • 34.34 The Michelson interferometer is used in a class of commercially available optical instruments called wavelength meters. In a wavelength meter, the interferometer is illuminated simultaneously with the parallel beam of a reference laser of known wavelength and that of an unknown laser. The movable mirror of the interferometer is then displaced by a distance Δd, and the number of fringes produced by each laser and passing by a reference point (a photo detector) is counted. In a given wavelength meter, a red He-Ne laser (λRed = 632.8 nm) is used as a reference laser. When the movable mirror of the interferometer is displaced by a distance Δd, a number ΔNRed = 6.000 · 104 red fringes and ΔNunknown = 7.780 · 104 fringes pass by the reference photodiode.

    1. Calculate the wavelength of the unknown laser.

    2. Calculate the displacement, Δd, of the movable mirror.

  • 34.35 Monochromatic blue light (λ = 449 nm) is beamed into a Michelson interferometer. How many fringes move by the screen when the movable mirror is moved a distance d = 0.381 mm?

    Answer

  • 34.36 At the Long-baseline Interferometer Gravitational-wave Observatory (LIGO) facilities in Hanford, Washington, and Livingston, Louisiana, laser beams of wavelength 550.0 nm travel along perpendicular paths 4.000 km long. Each beam is reflected along its path and back 100 times before the beams are combined and compared. If a gravitational wave increases the length of one path and decreases the other, each by 1.000 part in 1021, what is the phase difference between the two beams as a result?

Sections 34.6 and 34.7
  • 34.37 Light of wavelength 653 nm illuminates a slit. If the angle between the first dark fringes on either side of the central maximum is 32.0°, What is the width of the slit?

    Answer

  • 34.38 An instructor uses light of wavelength 633 nm to create a diffraction pattern with a slit of width 0.135 mm. How far away from the slit must the instructor place the screen in order for the full width of the central maximum to be 5.00 cm?

  • 34.39 What is the largest slit width for which there are no minima when the wavelength of the incident light on the single slit is 600. nm?

    Answer

  • 34.40 Plane light waves are incident on a single slit of width 2.00 cm. The second dark fringe is observed at 43.0° from the central axis. What is the wavelength of the light?

Section 34.8
  • 34.41 The Large Binocular Telescope (LBT), on Mount Graham near Tucson, Arizona, has two primary mirrors. The mirrors are centered a distance of 14.4 m apart, thus improving the Rayleigh limit. What is the minimum angular resolution of the LBT for green light, λ = 550 nm?

    Answer

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  • 34.42 A canvas tent has a single, tiny hole in its side. On the opposite wall of the tent, 2.0 m away, you observe a dot (due to the Sun's light incident upon the hole) of width 2.0 mm, with a faint ring around it. What is the size of the hole in the tent?

  • 34.43 Calculate and compare the angular resolutions of the Hubble Space Telescope (aperture diameter 2.40 m, wavelength 450. nm; illustrated in the text), the Keck Telescope (aperture diameter 10.0 m, wavelength 450. nm), and the Arecibo radio telescope (aperture diameter 305 m, wavelength 0.210 m). Assume that the resolution of each instrument is diffraction limited.

    Answer

  • 34.44 The Hubble Space Telescope (Figure 34.33) is capable of resolving optical images to an angular resolution of 2.80 · 10−7 rad with its 2.40-m mirror. How large would a radio telescope have to be in order to image an object in the radio spectrum with the same resolution, assuming the wavelength of the waves is 10.0 cm?

  • 34.45 Think of the pupil of your eye as a circular aperture 5.00 mm in diameter. Assume you are viewing light of wavelength 550 nm, to which your eyes are maximally sensitive.

    1. What is the minimum angular separation at which you can distinguish two stars?

    2. What is the maximum distance at which you can distinguish the two headlights of a car mounted 1.50 m apart?

    Answer

Section 34.9
  • 34.46 A red laser pointer with a wavelength of 635 nm is shined on a double slit producing a diffraction pattern on a screen that is 1.60 m behind the double slit. The central maximum of the diffraction pattern has a width of 4.20 cm, and the fourth bright spot is missing on both sides. What is the size of the individual slits, and what is the separation between them?

  • •34.47 A double slit is opposite the center of a 1.8-m wide screen 2.0 m from the slits. The slit separation is 24 μm and the width of each slit is 7.2 μm. How many fringes are visible on the screen if the slit is illuminated by 600.-nm light?

    Answer

  • 34.48 A two-slit apparatus is covered with a red (670 nm) filter. When white light is shone on the filter, on the screen beyond the two-slit apparatus, there are nine interference maxima within the 4.50-cm-wide central diffraction maximum. When a blue (450 nm) filter replaces the red, how many interference maxima will there be in the central diffraction maximum, and how wide will that diffraction maximum be?

  • 34.49 The irradiance pattern observed in a two-slit interference-diffraction experiment is presented in the figure. The red line represents the actual intensity measured as a function of angle, while the green line represents the envelope of the interference patterns.

    1. Determine the slit width a in terms of the wavelength λ of the light used in the experiment.

    2. Determine the center-to-center slit separation d in terms of the wavelength λ.

    3. Using the information in the graph, determine the ratio of slit width a to the center-to-center separation between the slits, d.

    4. Can you calculate the wavelength of light, actual slit separation, and slit width?

    Answer

Section 34.10
  • 34.50 Two different wavelengths of light are incident on a diffraction grating. One wavelength is 600. nm and the other is unknown. If the 3rd order of the unknown wavelength appears at the same position as the 2nd order of the 600. nm light, what is the value of the unknown wavelength?

  • 34.51 Light from an argon laser strikes a diffraction grating that has 7020 grooves per centimeter. The central and first-order principal maxima are separated by 0.332 m on a wall 1.00 m from the grating. Determine the wavelength of the laser light.

    Answer

  • 34.52 A 5.000-cm-wide diffraction grating with 200 grooves is used to resolve two closely spaced lines (a doublet) in a spectrum. The doublet consists of two wavelengths, λa = 629.8 nm and λb = 630.2 nm. The light illuminates the entire grating at normal incidence. Calculate to four significant digits the angles θ1a and θ1b with respect to the normal at which the first-order diffracted beams for the two wavelengths, λa and λb, respectively, will be reflected from the grating. Note that this is not 0°! What order of diffraction is required to resolve these two lines using this grating?

  • 34.53 A diffraction grating has 4.00 · 103 lines/cm and has white light (400.−700. nm) incident on it. What wavelength(s) will be visible at 45.0°?

    Answer

Section 34.11
  • 34.54 What is the wavelength of the X-rays if the first-order Bragg diffraction is observed at 23.0° related to the crystal surface, with inter atomic distance of 0.256 nm?

Additional Problems
  • 34.55 How many lines per centimeter must a grating have if there is to be no second-order spectrum for any visible wavelength (400−750 nm)?

    Answer

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  • 34.56 Many times, radio antennas occur in pairs. The effect is that they will then produce constructive interference in one direction while producing destructive interference in another direction—a directional antenna—so that their emissions don't overlap with nearby stations. How far apart at a minimum should a local radio station, operating at 88.1 MHz, place its pair of antennae operating in phase such that no emission occurs along a line 45.0° from the line joining the antennae?

  • 34.57 A laser produces a coherent beam of light that does not spread (diffract) as much in comparison to other light sources like an incandescent bulb. Lasers therefore have been used for measuring large distances, such as the distance between the Moon and the Earth with very great accuracy. In one such experiment, a laser pulse (wavelength 633 nm) is fired at the Moon. What should be the size of the circular aperture of the laser source that would produce central maximum of 1.00-km diameter on the surface of the Moon? Distance between the Moon and the Earth is 3.84 · 105 km.

    Answer

  • 34.58 A diffraction grating with exactly 1000 lines per centimeter is illuminated by a He-Ne laser of wavelength 633 nm.

    1. What is the highest order of diffraction that could be observed with this grating?

    2. What would be the highest order if there were exactly 10,000 lines per centimeter?

  • 34.59 The thermal stability of a Michelson interferometer can be improved by submerging it in water. Consider an interferometer that is submerged in water, measuring light from a monochromatic source that is in air. If the movable mirror moves a distance of d = 0.200 mm, exactly N = 800 fringes move by the screen. What is the original wavelength (in air) of the monochromatic light?

    Answer

  • 34.60 A Blu-ray disc uses a blue laser with a free-space wavelength of 405 nm. If the disc is protected with polycarbonate (n = 1.58), determine the minimum thickness of the disc for destructive interference. Compare this value to that for CDs illuminated by infrared light.

  • 34.61 An airplane is made invisible to radar by coating it with a 5.00-mm-thick layer of an antireflective polymer with the index of refraction n = 1.50. What is the wavelength of radar waves for which the plane is made invisible?

    Answer

  • 34.62 Coherent monochromatic light passes through parallel slits and then onto a screen that is at a distance L = 2.40 m from the slits. The narrow slits are a distance d = 2.00 · 10−5 m apart. If the minimum spacing between bright spots is y = 6.00 cm, find the wavelength of the light.

  • 34.63 Determine the minimum thickness of a soap film (n = 1.32) that would produce constructive interference when illuminated by light of wavelength of 550. nm.

    Answer

  • 34.64 You are making a diffraction grating that is required to separate the two spectral lines in the sodium D doublet, at wavelengths 588.9950 and 589.5924 nm, by at least 2.00 mm on a screen that is 80.0 cm from the grating. The lines are to be ruled over a distance of 1.50 cm on the grating. What is the minimum number of lines you should have on the grating?

  • 34.65 A Michelson interferometer is illuminated with a 600.-nm light source. How many fringes are observed if one of the mirrors of the interferometer is moved a distance of 200. μm?

    Answer

  • 34.66 What is the smallest object separation you can resolve with your naked eye? Assume the diameter of your pupil is 3.5 mm, and that your eye has a near point of 25 cm and a far point of infinity.

  • 34.67 When using a telescope with an objective of diameter 12.0 cm, how close can two features on the Moon be and still be resolved? Take the wavelength of light to be 550 nm, near the center of the visible spectrum.

    Answer

  • 34.68 There is air on both sides of a soap film. What is the smallest thickness of the soap film (n = 1.420) that would appear dark if illuminated with 500.-nm light?

  • 34.69 X-rays with a wavelength of 1.00 nm are scattered off of two small tumors in the human body. If the two tumors are a distance of 10.0 cm away from the X-ray detector, which has an entrance aperture of 1.00 mm, what is the minimum separation between the two tumors that will allow the X-ray detector to determine that there are two tumors instead of one?

    Answer

  • 34.70 A glass with a refractive index of 1.50 is inserted into one arm of a Michelson interferometer that uses a 600.-nm light source. This causes the fringe pattern to shift by exactly 1000 fringes. How thick is the glass?

  • •34.71 White light is shone on a very thin layer of mica (n = 1.57), and above the mica layer, interference maxima for two wavelengths (and no other in between) are seen: one blue wavelength of 480 nm, and one yellow wavelength of 560 nm. What is the thickness of the mica layer?

    Answer

  • 34.72 In a double-slit arrangement the slits are 1.00 · 10−5 m apart. If light with wavelength 500. nm passes through the slits, what will be the distance between the m = 1 and m = 3 maxima on a screen 1.00 m away?

  • 34.73 A Newton's ring apparatus consists of a convex lens with a large radius of curvature R placed on a flat glass disc. (a) Show that the distance x from the center to the air, thickness d, and the radius of curvature R are given by x2 = 2Rd. (b) Show that the radius of nth constructive interference is given by . (c) How many bright fringes may be seen if it is viewed by red light of wavelength 700. nm for R = 10.0 m, and the plane glass disc diameter is 5.00 cm?

    Answer