4 The Diffraction Grating

Section 38.4 The Diffraction Grating

Compact disc

Note: In the following problems, assume that the light is incident normally on the gratings.

23. White light is spread out into its spectral components by

a diffraction grating. If the grating has 2 000 grooves per centimeter, at what angle does red light of wavelength

Central 640 nm appear in first order? maximum

24. Diffraction Light from an argon laser strikes a diffraction grating that

grating has 5 310 grooves per centimeter. The central and first-

First-order

maxima

order principal maxima are separated by 0.488 m on a wall

1.72 m from the grating. Determine the wavelength of the laser light.

25. The hydrogen spectrum has a red line at 656 nm and

a blue line at 434 nm. What are the angular separations between these two spectral lines obtained with a diffraction

Laser

grating that has 4 500 grooves/cm?

Figure P38.30

26. A helium–neon laser (% " 632.8 nm) is used to calibrate a diffraction grating. If the first-order maximum occurs at 20.5°, what is the spacing between adjacent grooves in the

the desired location. Assume that the laser light has a grating?

wavelength of 780 nm and that the diffraction grating is

27. Three discrete spectral lines occur at angles of 10.09°, positioned 6.90 /m from the disk. Assume that the 13.71°, and 14.77° in the first-order spectrum of a grating

first-order beams are to fall on the disk 0.400 /m on either spectrometer. (a) If the grating has 3 660 slits/cm, what

side of the information track. What should be the number are the wavelengths of the light? (b) At what angles are

of grooves per millimeter in the grating? these lines found in the second-order spectrum?

31. A source emits 531.62-nm and 531.81-nm light. (a) What

28. Show that, whenever white light is passed through a diffrac- minimum number of grooves is required for a grating that tion grating of any spacing size, the violet end of the contin-

resolves the two wavelengths in the first-order spectrum? uous visible spectrum in third order always overlaps with red

(b) Determine the slit spacing for a grating 1.32 cm wide light at the other end of the second-order spectrum.

that has the required minimum number of grooves.

32. A diffraction grating has 4 200 rulings/cm. On a screen with 3 000 grooves/cm. (a) What is the resolving power of

29. A diffraction grating of width 4.00 cm has been ruled

2.00 m from the grating, it is found that for a particular this grating in the first three orders? (b) If two monochro-

order m, the maxima corresponding to two closely spaced matic waves incident on this grating have a mean

wavelengths of sodium (589.0 nm and 589.6 nm) are sepa- wavelength of 400 nm, what is their wavelength separation

rated by 1.59 mm. Determine the value of m. if they are just resolved in the third order?

33. A grating with 250 grooves/mm is used with an incandes-

30. The laser in a CD player must precisely follow the spiral cent light source. Assume the visible spectrum to range in track, along which the distance between one loop of the

wavelength from 400 to 700 nm. In how many orders can spiral and the next is only about 1.25 /m. A feedback

one see (a) the entire visible spectrum and (b) the short- mechanism lets the player know if the laser drifts off the

wavelength region?

track, so that the player can steer it back again. Figure

34. A wide beam of laser light with a wavelength of 632.8 nm is P38.30 shows how a diffraction grating is used to provide

directed through several narrow parallel slits, separated by information to keep the beam on track. The laser light

1.20 mm, and falls on a sheet of photographic film 1.40 m passes through a diffraction grating just before it reaches

away. The exposure time is chosen so that the film stays the disk. The strong central maximum of the diffraction

unexposed everywhere except at the central region of each pattern is used to read the information in the track of pits.

bright fringe. (a) Find the distance between these interfer- The two first-order side maxima are used for steering. The

ence maxima. The film is printed as a transparency—it is grating is designed so that the first-order maxima fall on

opaque everywhere except at the exposed lines. Next, the the flat surfaces on both sides of the information track.

same beam of laser light is directed through the trans- Both side beams are reflected into their own detectors. As

parency and allowed to fall on a screen 1.40 m beyond. long as both beams are reflecting from smooth nonpitted

(b) Argue that several narrow parallel bright regions, sepa- surfaces, they are detected with constant high intensity. If

rated by 1.20 mm, will appear on the screen, as real images the main beam wanders off the track, however, one of the

of the original slits. If at last the screen is removed, light will side beams will begin to strike pits on the information

diverge from the images of the original slits with the same track and the reflected light will diminish. This change is

reconstructed wave fronts as the original slits produced. used with an electronic circuit to guide the beam back to

(Suggestion: You may find it useful to draw diagrams similar

Problems

to Figure 38.16. A train of thought like this, at a soccer common vertical direction. A plane-polarized beam of game, led Dennis Gabor to the invention of holography.)

light with E 0 parallel to the vertical reference direction is incident from the left on the first disk with intensity

10.0 units (arbitrary). Calculate the transmitted inten-

Section 38.5 Diffraction of X-Rays by Crystals

sity I f when (a) ! 1 " 20.0°, ! 2 " 40.0°, and ! 3 " 60.0°;

35. Potassium iodide (KI) has the same crystalline structure as (b) ! 1 " 0°, ! 2 " 30.0°, and ! 3 " 60.0°. NaCl, with atomic planes separated by 0.353 nm. A mono-

43. The angle of incidence of a light beam onto a reflecting chromatic x-ray beam shows a first-order diffraction maxi-

surface is continuously variable. The reflected ray is found mum when the grazing angle is 7.60°. Calculate the x-ray

to be completely polarized when the angle of incidence is wavelength.

48.0°. What is the index of refraction of the reflecting

36. A wavelength of 0.129 nm characterizes K 2 x-rays from

material?

zinc. When a beam of these x-rays is incident on the

44. Review Problem. (a) A transparent plate with index of re- surface of a crystal whose structure is similar to that of

fraction n 2 is immersed in a medium with index n 1 . Light NaCl, a first-order maximum is observed at 8.15°. Calculate

traveling in the surrounding medium strikes the top surface the interplanar spacing based on this information.

of the plate at Brewster’s angle. Show that if and only if the

37. If the interplanar spacing of NaCl is 0.281 nm, what is surfaces of the plate are parallel, the refracted light will the predicted angle at which 0.140-nm x-rays are diffracted

strike the bottom surface of the plate at Brewster’s angle in a first-order maximum?

for that interface. (b) What If? Instead of a plate, consider

a prism of index of refraction n 2 separating media of dif-

38. The first-order diffraction maximum is observed at 12.6° ferent refractive indices n 1 and n 3 . Is there one particular for a crystal in which the interplanar spacing is 0.240 nm.

apex angle between the surfaces of the prism for which How many other orders can be observed?

light can fall on both of its surfaces at Brewster’s angle as it

39. In water of uniform depth, a wide pier is supported on passes through the prism? If so, determine it. pilings in several parallel rows 2.80 m apart. Ocean waves of uniform wavelength roll in, moving in a direction that

45. The critical angle for total internal reflection for sapphire makes an angle of 80.0° with the rows of posts. Find the

surrounded by air is 34.4°. Calculate the polarizing angle three longest wavelengths of waves that will be strongly

for sapphire.

reflected by the pilings.

46. For a particular transparent medium surrounded by air, show that the critical angle for total internal reflection and

Section 38.6 Polarization of Light Waves

the polarizing angle are related by cot ! p " sin ! c .

47. How far above the horizon is the Moon when its image Problem 34 in Chapter 34 can be assigned with this

reflected in calm water is completely polarized? (n water " section.

40. Unpolarized light passes through two polaroid sheets. The

Additional Problems

axis of the first is vertical, and that of the second is at 30.0°

48. In Figure P38.42, suppose that the transmission axes of the to the vertical. What fraction of the incident light is

left and right polarizing disks are perpendicular to each transmitted?

other. Also, let the center disk be rotated on the common

41. Plane-polarized light is incident on a single polarizing disk axis with an angular speed 3. Show that if unpolarized

light is incident on the left disk with an intensity I max , the transmission axis. Through what angle should the disk be

with the direction of E 0 parallel to the direction of the

intensity of the beam emerging from the right disk is rotated so that the intensity in the transmitted beam is

I" reduced by a factor of (a) 3.00, (b) 5.00, (c) 10.0? 1 16 I max (1 ' cos 43t)

42. Three polarizing disks whose planes are parallel are cen- This means that the intensity of the emerging beam is tered on a common axis. The direction of the transmission

modulated at a rate that is four times the rate of rotation axis in each case is shown in Figure P38.42 relative to the

of the center disk. [Suggestion: Use the trigonometric iden- tities cos 2 ! " (1 , cos 2!)/2 and sin 2 ! " (1 ' cos 2!)/2,

θ 1 and recall that ! " 3t.] θ 2 49. You want to rotate the plane of polarization of a polarized

light beam by 45.0° with a maximum intensity reduction of

I i θ 3 10.0%. (a) How many sheets of perfect polarizers do you need to achieve your goal? (b) What is the angle between adjacent polarizers?

50. Figure P38.50 shows a megaphone in use. Construct a the-

I f oretical description of how a megaphone works. You may Figure P38.42 Problems 42 and 48.

assume that the sound of your voice radiates just through

CHAPTER 38• Diffraction Patterns and Polarization

angular resolution of the hawk’s eye. Assume that the hawk is most sensitive to green light having a wavelength of 500 nm and that it has a pupil of diameter 12.0 mm. (d) A mouse is on the ground 30.0 m below. By what distance must the mouse’s whiskers be separated if the hawk can re- solve them?

55. A 750-nm light beam hits the flat surface of a certain liquid, and the beam is split into a reflected ray and a refracted ray. If the reflected ray is completely polarized at 36.0°, what is the wavelength of the refracted ray?

Susan Allen Sigmon/Getty Images

56. Iridescent peacock feathers are shown in Figure P38.56a.

Figure P38.50

The surface of one microscopic barbule is composed of transparent keratin that supports rods of dark brown melanin in a regular lattice, represented in Figure P38.56b.

the opening of your mouth. Most of the information in (Your fingernails are made of keratin, and melanin is the speech is carried not in a signal at the fundamental fre-

dark pigment giving color to human skin.) In a portion of quency, but in noises and in harmonics, with frequencies

the feather that can appear turquoise, assume that the of a few thousand hertz. Does your theory allow any pre-

melanin rods are uniformly separated by 0.25 /m, with air diction that is simple to test?

between them. (a) Explain how this structure can appear

51. Light from a helium–neon laser (% " 632.8 nm) is inci- blue-green when it contains no blue or green pigment. dent on a single slit. What is the maximum width of the slit for which no diffraction minima are observed?

52. What are the approximate dimensions of the smallest object on Earth that astronauts can resolve by eye when they are orbiting 250 km above the Earth? Assume that % " 500 nm and that a pupil diameter is 5.00 mm.

53. Review problem.

A beam of 541-nm light is incident on a dif- fraction grating that has 400 grooves/mm. (a) Determine the angle of the second-order ray. (b) What If? If the entire apparatus is immersed in water, what is the new second-order angle of diffraction? (c) Show that the two diffracted rays of parts (a) and (b) are related through the law of refraction.

54. The Very Large Array (VLA) is a set of 27 radio telescope dishes in Caton and Socorro Counties, New Mexico (Fig. P38.54). The antennas can be moved apart on railroad tracks, and their combined signals give the resolving power

© Diane Schiumo 1988/Fundamental Photographs

of a synthetic aperture 36.0 km in diameter. (a) If the detectors are tuned to a frequency of 1.40 GHz, what is the Photo

(a)

angular resolution of the VLA? (b) Clouds of hydrogen radiate at this frequency. What must be the separation distance of two clouds 26 000 lightyears away at the center of the galaxy, if they are to be resolved? (c) What If? As the telescope looks up, a circling hawk looks down. Find the

(b)

Figure P38.56 (a) Iridescent peacock feathers. (b) Microscopic Figure P38.54 Some of the radio telescope dishes in the Very

Riccardo Giovanelli and Martha Haynes, Cornell University

section of a feather showing dark melanin rods in a pale keratin

Problems

(b) Explain how it can also appear violet if light falls on it that the average wavelength of the light coming from the in a different direction. (c) Explain how it can present

screen is 550 nm.

different colors to your two eyes at the same time—a char-

62. (a) Light traveling in a medium of index of refraction n 1 is acteristic of iridescence. (d) A compact disc can appear to

incident at an angle ! on the surface of a medium of index

be any color of the rainbow. Explain why this portion of the n 2 . The angle between reflected and refracted rays is ). feather cannot appear yellow or red. (e) What could be

Show that

different about the array of melanin rods in a portion of the feather that does appear to be red?

n sin )

tan ! "

57. Light of wavelength 500 nm is incident normally on a dif- n 1 'n 2 cos ) fraction grating. If the third-order maximum of the diffrac-

(Suggestion: Use the identity sin(A , B) " sin A cos B , tion pattern is observed at 32.0°, (a) what is the number of

cos A sin B.) (b) What If ? Show that this expression for tan ! rulings per centimeter for the grating? (b) Determine the

1, and n 2 " n. total number of primary maxima that can be observed in

reduces to Brewster’s law when ) " 90°, n 1 "

this situation.

63. Suppose that the single slit in Figure 38.6 is 6.00 cm wide and in front of a microwave source operating at 7.50 GHz.

58. Light strikes a water surface at the polarizing angle. The (a) Calculate the angle subtended by the first minimum in part of the beam refracted into the water strikes a sub-

the diffraction pattern. (b) What is the relative intensity merged glass slab (index of refraction, 1.50), as shown in

I/I max at ! " 15.0°? (c) Assume that two such sources, sep- Figure P38.58. The light reflected from the upper surface

arated laterally by 20.0 cm, are behind the slit. What must of the slab is completely polarized. Find the angle between

the maximum distance between the plane of the sources the water surface and the glass slab.

and the slit be if the diffraction patterns are to be resolved? (In this case, the approximation sin ! ! tan ! is not valid because of the relatively small value of a/%.)

64. Two polarizing sheets are placed together with their trans- mission axes crossed so that no light is transmitted. A third

θ sheet is inserted between them with its transmission axis at

Air

Water

an angle of 45.0° with respect to each of the other axes. Find the fraction of incident unpolarized light intensity transmitted by the three-sheet combination. (Assume each polarizing sheet is ideal.)

65. Two wavelengths % and % , (% (with (% ++ %) are inci-

Figure P38.58

dent on a diffraction grating. Show that the angular sepa- ration between the spectral lines in the mth-order spectrum is

59. A beam of bright red light of wavelength 654 nm passes through a diffraction grating. Enclosing the space beyond

the grating is a large screen forming one half of a cylinder

(d/m) √ 2 ' % 2

centered on the grating, with its axis parallel to the slits in the grating. Fifteen bright spots appear on the screen.

where d is the slit spacing and m is the order number. Find the maximum and minimum possible values for the

66. Two closely spaced wavelengths of light are incident on a slit separation in the diffraction grating.

diffraction grating. (a) Starting with Equation 38.10, show

60. A pinhole camera has a small circular aperture of diameter D. that the angular dispersion of the grating is given by Light from distant objects passes through the aperture into

d!

an otherwise dark box, falling on a screen located a distance

L away. If D is too large, the display on the screen will be

d cos ! fuzzy, because a bright point in the field of view will send

(b) A square grating 2.00 cm on each side containing light onto a circle of diameter slightly larger than D. On the

8 000 equally spaced slits is used to analyze the spectrum of other hand, if D is too small, diffraction will blur the display

mercury. Two closely spaced lines emitted by this element on the screen. The screen shows a reasonably sharp image if

have wavelengths of 579.065 nm and 576.959 nm. What is the diameter of the central disk of the diffraction pattern,

the angular separation of these two wavelengths in the specified by Equation 38.9, is equal to D at the screen.

second-order spectrum?

(a) Show that for monochromatic light with plane wave

67. The scale of a map is a number of kilometers per centime- fronts and L 44 D, the condition for a sharp view is fulfilled

ter, specifying the distance on the ground that any dis- if D 2 " 2.44%L. (b) Find the optimum pinhole diameter for

tance on the map represents. The scale of a spectrum is its 500-nm light projected onto a screen 15.0 cm away.

dispersion, a number of nanometers per centimeter, which

61. An American standard television picture is composed of specifies the change in wavelength that a distance across about 485 horizontal lines of varying light intensity.

the spectrum represents. One must know the dispersion in Assume that your ability to resolve the lines is limited only

order to compare one spectrum with another and to make by the Rayleigh criterion and that the pupils of your eyes

a measurement of (for example) a Doppler shift. Let y rep- are 5.00 mm in diameter. Calculate the ratio of minimum

resent the position relative to the center of a diffraction viewing distance to the vertical dimension of the picture

pattern projected onto a flat screen at distance L by a dif-

CHAPTER 38• Diffraction Patterns and Polarization

(a) Prove that the dispersion is given by maximum of the central maximum of the single-slit

Fraunhofer diffraction pattern. You can evaluate this angle

" dy

of spreading in this problem and in the next. (a) In Equa- tion 38.4, define )/2 " 5 and show that, at the point

m(L 2 ,y 2 ) 3/2

where I " 0.5I max , we must have sin 5 " 5/ √ 2 . (b) Let

(b) Calculate the dispersion in first order for light with a

y 1 " sin 5 and y 2 " 5/ √ 2 . Plot y 1 and y 2 on the same set

mean wavelength of 550 nm, analyzed with a grating having

8 000 rulings/cm, and projected onto a screen 2.40 m away. of axes over a range from 5 " 1 rad to 5 " */2 rad. De-

68. Derive Equation 38.12 for the resolving power of a grating, termine 5 from the point of intersection of the two curves. R " Nm, where N is the number of slits illuminated and m

(c) Then show that, if the fraction %/a is not large, the an- is the order in the diffraction pattern. Remember that

gular full width at half maximum of the central diffraction Rayleigh’s criterion (Section 38.3) states that two

maximum is (! " 0.886 %/a.

71. Another method to solve the equation 5 " √ 2 sin 5 in

wavelengths will be resolved when the principal maximum

for one falls on the first minimum for the other. Problem 70 is to guess a first value of 5, use a computer or

69. Figure P38.69a is a three-dimensional sketch of a birefrin- calculator to see how nearly it fits, and continue to update gent crystal. The dotted lines illustrate how a thin parallel-

your estimate until the equation balances. How many steps faced slab of material could be cut from the larger specimen

(iterations) does this take?

with the optic axis of the crystal parallel to the faces of the

72. In the diffraction pattern of a single slit, described by plate. A section cut from the crystal in this manner is known

the equation

as a retardation plate. When a beam of light is incident on the

I ! "I max ()/2) ()/2) shown in Figure P38.69b, the O ray and the E ray travel 2

sin 2

plate perpendicular to the direction of the optic axis, as

along a single straight line, but with different speeds. (a) Let with ) " (2*a sin !)/%, the central maximum is at ) " 0 the thickness of the plate be d and show that the phase dif-

and the side maxima are approximately at )/2 " (m , )* 1 2 ference between the O ray and the E ray is

for m " 1, 2, 3, . . . . Determine more precisely (a) the location of the first side maximum, where m " 1, and

(b) the location of the second side maximum. Observe in

Figure 38.10a that the graph of intensity versus )/2 has a where % is the wavelength in air. (b) In a particular case

horizontal tangent at maxima and also at minima. You will the incident light has a wavelength of 550 nm. Find the

need to solve a transcendental equation. minimum value of d for a quartz plate for which ! " */2.

73. Light of wavelength 632.8 nm illuminates a single slit, Such a plate is called a quarter-wave plate. (Use values of n O

and a diffraction pattern is formed on a screen 1.00 m from and n E from Table 38.1.)

the slit. Using the data in the table below, plot relative inten- sity versus position. Choose an appropriate value for the slit width a and on the same graph used for the experimental data, plot the theoretical expression for the relative intensity

sin 2 ()/2)

I max ()/2) 2 Optic

What value of a gives the best fit of theory and experiment? axis

Position Relative to Relative Intensity Central Maximum (mm)

Wave front

Figure P38.69

12.9 undergo? One quantitative answer is the full width at half

70. How much diffraction spreading does a light beam

Answers to Quick Quizzes

1241

0.012

14.5 aperture larger, relative to the light wavelength, increasing 0.016

15.3 the resolving power. Thus, we should choose a blue filter. 0.015

16.1 38.7 (b). The tracks of information on a compact disc are 0.010

16.9 much closer together than on a phonograph record. As 0.004 4

17.7 a result, the diffraction maxima from the compact disc 0.000 6

18.5 will be farther apart than those from the record. 0.000 3

19.3

0.003

20.2 38.8 (c). With the doubled wavelength, the pattern will be wider. Choices (a) and (d) make the pattern even wider.

From Equation 38.10, we see that choice (b) causes sin !

Answers to Quick Quizzes

to be twice as large. Because we cannot use the small

38.1 (a). Equation 38.1 shows that a decrease in a results in an angle approximation, however, a doubling of sin ! is not increase in the angles at which the dark fringes appear.

the same as a doubling of !, which would translate to a

38.2 (c). The space between the slightly open door and the doubling of the position of a maximum along the screen. doorframe acts as a single slit. Sound waves have

If we only consider small-angle maxima, choice (b) would wavelengths that are larger than the opening and so

work, but it does not work in the large-angle case. are diffracted and spread throughout the room you are

38.9 (b). Electric field vectors parallel to the metal wires cause in. Because light wavelengths are much smaller than the

electrons in the metal to oscillate parallel to the wires. slit width, they experience negligible diffraction. As a

Thus, the energy from the waves with these electric field result, you must have a direct line of sight to detect the

vectors is transferred to the metal by accelerating these light waves.

electrons and is eventually transformed to internal energy

38.3 The situation is like that depicted in Figure 38.11 except through the resistance of the metal. Waves with electric- that now the slits are only half as far apart. The diffrac-

field vectors perpendicular to the metal wires pass tion pattern is the same, but the interference pattern is

through because they are not able to accelerate electrons stretched out because d is smaller. Because d/a " 3, the

in the wires.

m " 3 interference maximum coincides with the first

38.10 (c). At some intermediate distance, the light rays from diffraction minimum. Your sketch should look like the

the fixtures will strike the floor at Brewster’s angle and figure below.

reflect to your eyes. Because this light is polarized hori- zontally, it will not pass through your polarized sun-

I glasses. Tilting your head to the side will cause the reflections to reappear.

β /2

38.4 (c). In Equation 38.7, the ratio d/a is independent of wavelength, so the number of interference fringes in the central diffraction pattern peak remains the same. Equa- tion 38.1 tells us that a decrease in wavelength causes a decrease in the width of the central peak.

38.5 (b). The effective slit width in the vertical direction of the cat’s eye is larger than that in the horizontal direc- tion. Thus, the eye has more resolving power for lights separated in the vertical direction and would be more effective at resolving the mast lights on the boat.

38.6 (a). We would like to reduce the minimum angular sepa- ration for two objects below the angle subtended by the two stars in the binary system. We can do that by reducing

the wavelength of the light—this in essence makes the 2003 by Sidney Harris