Physics For Scientists And Engineers 6E - part 305

S E C T I O N 3 8 . 4 • The Diffraction Grating

1217

min

Figure 38.14 (Example 38.4) Two point sources separated by a

distance d as observed by the eye.

We can use this result to determine the minimum sepa-

ration distance d between two point sources that the eye can
distinguish if they are a distance L from the observer (Fig.
38.14). Because !

min

is small, we see that

For example, if the point sources are 25 cm from the eye
(the near point), then

This is approximately equal to the thickness of a human hair.

d " (25 cm)(3 & 10

rad) " 8 & 10

d " L!

min

sin !

min

! !

min

Example 38.5 Resolution of a Telescope

The Keck telescope at Mauna Kea, Hawaii, has an effective
diameter of 10 m. What is its limiting angle of resolution for
600-nm light?

Solution Because D " 10 m and % " 6.00 & 10

m, Equa-

tion 38.9 gives

Any two stars that subtend an angle greater than or equal to
this value are resolved (if atmospheric conditions are ideal).

The Keck telescope can never reach its diffraction limit

because the limiting angle of resolution is always set by atmos-
pheric blurring at optical wavelengths. This seeing limit is usu-
ally about 1 s of arc and is never smaller than about 0.1 s of
arc. (This is one of the reasons for the superiority of pho-
tographs from the Hubble Space Telescope, which views celes-
tial objects from an orbital position above the atmosphere.)

What If?

What if we consider radio telescopes? These

are much larger in diameter than optical telescopes, but

0.015 s of arc

7.3 & 10

rad

min

1.22

6.00 & 10

10 m

do they  have  angular  resolutions  that  are  better  than
optical  telescopes?  For  example,  the  radio  telescope  at
Arecibo,  Puerto  Rico,  has  a  diameter  of  305 m  and  is
designed  to  detect  radio  waves  of  0.75-m  wavelength.
How does  its  resolution  compare  to  that  of  the  Keck
telescope?

Answer The increase in diameter might suggest that radio
telescopes would have better resolution, but Equation 38.9
shows that !

min

depends on both diameter and wavelength.

Calculating the minimum angle of resolution for the radio
telescope, we find

Notice that this limiting angle of resolution is measured in
minutes of arc rather than the seconds of arc for the optical
telescope.  Thus,  the  change  in  wavelength  more  than
compensates for the increase in diameter, and the limiting
angle  of  resolution  for  the  Arecibo  radio  telescope  is
more  than  40 000  times  larger  (that  is,  worse)  than  the
Keck minimum.

" 3.0 & 10

rad

! 10 min of arc

min

1.22

0.75 m

305 m

As an example of the effects of atmospheric blurring mentioned in Example 38.5, con-
sider telescopic images of Pluto and its moon Charon. Figure 38.15a shows the image
taken in 1978 that represents the discovery of Charon. In this photograph taken from
an Earth-based telescope, atmospheric turbulence causes the image of Charon to
appear only as a bump on the edge of Pluto. In comparison, Figure 38.15b shows a
photograph taken with the Hubble Space Telescope. Without the problems of atmos-
pheric turbulence, Pluto and its moon are clearly resolved.

38.4 The Diffraction Grating

The

diffraction grating, a useful device for analyzing light sources, consists of a large

number of equally spaced parallel slits. A transmission grating can be made by cutting par-
allel grooves on a glass plate with a precision ruling machine. The spaces between the
grooves are transparent to the light and hence act as separate slits. A reflection grating can

be made by cutting parallel grooves on the surface of a reflective material. The reflection
of light from the spaces between the grooves is specular, and the reflection from the
grooves cut into the material is diffuse. Thus, the spaces between the grooves act as paral-
lel sources of reflected light, like the slits in a transmission grating. Current technology
can produce gratings that have very small slit spacings. For example, a typical grating
ruled with 5 000 grooves/cm has a slit spacing d " (1/5 000) cm " 2.00 & 10

cm.

A section of a diffraction grating is illustrated in Figure 38.16. A plane wave is inci-

dent from the left, normal to the plane of the grating. The pattern observed on the

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C H A P T E R 3 8 • Diffraction Patterns and Polarization

(a)

Pluto

Charon

(b)

Figure 38.15 (a) The photograph on which Charon, the moon of Pluto, was discov-

ered in 1978. From an Earth-based telescope, atmospheric blurring results in Charon

appearing only as a subtle bump on the edge of Pluto. (b) A Hubble Space Telescope

photo of Pluto and Charon, clearly resolving the two objects.

Photo courtesy of Gemini Observatory

U.S. Naval Observatory/James W

. Christy

, U.S. Navy photograph

= d sin θ

First-order

maximum

(m = 1)

Central or

zeroth-order

maximum

(m = 0)

First-order

maximum

(m = –1)

Incoming plane

wave of light

Diffraction

grating

Figure 38.16 Side view of a diffraction grating. The slit separation is d, and the path

difference between adjacent slits is d sin

▲

PITFALL PREVENTION

38.3 A Diffraction Grating

Is an Interference
Grating

As  with  diffraction  pattern,  diffrac-
tion  grating is  a  misnomer,  but
is deeply  entrenched  in  the
language  of  physics.  The  diffrac-
tion  grating  depends  on  diffrac-
tion  in  the  same  way  as  the
double  slit—spreading  the  light
so  that  light  from  different  slits
can  interfere.  It  would  be  more
correct  to  call  it  an  interference
grating, but  diffraction  grating is
the name in use.

screen (far to the right of Figure 38.16) is the result of the combined effects of inter-
ference and diffraction. Each slit produces diffraction, and the diffracted beams inter-
fere with one another to produce the final pattern.

The waves from all slits are in phase as they leave the slits. However, for some arbi-

trary direction ! measured from the horizontal, the waves must travel different path
lengths before reaching the screen. From Figure 38.16, note that the path difference 0
between rays from any two adjacent slits is equal to d sin !. If this path difference
equals one wavelength or some integral multiple of a wavelength, then waves from all
slits are in phase at the screen and a bright fringe is observed. Therefore, the condi-
tion for maxima in the interference pattern at the angle !

bright

(38.10)

We can use this expression to calculate the wavelength if we know the grating

spacing d and the angle !

bright

. If the incident radiation contains several wavelengths, the

mth-order maximum for each wavelength occurs at a specific angle. All wavelengths are
seen at ! " 0, corresponding to

m " 0, the zeroth-order maximum. The first-order maxi-

mum (m " 1) is observed at an angle that satisfies the relationship sin !

bright

/d; the

second-order maximum (m " 2) is observed at a larger angle !

bright

, and so on.

The intensity distribution for a diffraction grating obtained with the use of a mono-

chromatic source is shown in Figure 38.17. Note the sharpness of the principal maxima
and the broadness of the dark areas. This is in contrast to the broad bright fringes
characteristic of the two-slit interference pattern (see Fig. 37.7). You should also review
Figure 37.14, which shows that the width of the intensity maxima decreases as the
number of slits increases. Because the principal maxima are so sharp, they are much
brighter than two-slit interference maxima.

A schematic drawing of a simple apparatus used to measure angles in a diffraction

pattern is shown in Figure 38.18. This apparatus is a diffraction grating spectrometer. The
light to be analyzed passes through a slit, and a collimated beam of light is incident on
the grating. The diffracted light leaves the grating at angles that satisfy Equation 38.10,
and a telescope is used to view the image of the slit. The wavelength can be deter-
mined by measuring the precise angles at which the images of the slit appear for the
various orders.

m " 0, #1, #2, #3,

$ $ $

d sin !

bright

S E C T I O N 3 8 . 4 • The Diffraction Grating

1219

_ 2 _1 0

sin

Active Figure 38.17 Intensity

versus sin

for a diffraction grating.

The zeroth-, first-, and second-order

maxima are shown.

At the Active Figures link

at http://www.pse6.com, you

can choose the number of slits

to be illuminated to see the

effect on the interference

pattern.

Telescope

Slit

Source

Grating

Collimator

Active Figure 38.18 Diagram of a diffraction grating spectrometer. The collimated

beam incident on the grating is spread into its various wavelength components with

constructive interference for a particular wavelength occurring at the angles

bright

that

satisfy the equation d sin

bright

, where m " 0, 1, 2, . . . .

Use the spectrometer at

the Active Figures link at

http://www.pse6.com to observe

constructive interference for

various wavelengths.

Condition for interference

maxima for a grating

The spectrometer is a useful tool in atomic spectroscopy, in which the light from an

atom is analyzed to find the wavelength components. These wavelength components
can be used to identify the atom. We will investigate atomic spectra in Chapter 42 of
the extended version of this text.

Another application of diffraction gratings is in the recently developed grating light

valve (GLV), which may compete in the near future in video projection with the digital
micromirror devices (DMDs) discussed in Section 35.4. The grating light valve consists
of a silicon microchip fitted with an array of parallel silicon nitride ribbons coated with
a thin layer of aluminum (Fig. 38.19). Each ribbon is about 20 /m long and about
5 /m wide and is separated from the silicon substrate by an air gap on the order of
100 nm. With no voltage applied, all ribbons are at the same level. In this situation, the
array of ribbons acts as a flat surface, specularly reflecting incident light.

When a voltage is applied between a ribbon and the electrode on the silicon

substrate,  an  electric  force  pulls  the  ribbon  downward,  closer  to  the  substrate.
Alternate  ribbons  can  be  pulled  down,  while  those  in  between  remain  in  the  higher
configuration. As a result, the array of ribbons acts as a diffraction grating, such that
the  constructive  interference  for  a  particular  wavelength  of  light  can  be  directed
toward a screen or other optical display system. By using three such devices, one each
for red, blue, and green light, full-color display is possible.

The GLV tends to be simpler to fabricate and higher in resolution than compara-

ble DMD devices. On the other hand, DMD devices have already made an entry into
the market. It will be interesting to watch this technology competition in future years.

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C H A P T E R 3 8 • Diffraction Patterns and Polarization

Figure 38.19 A small portion of a

grating light valve. The alternating

reflective ribbons at different levels act

as a diffraction grating, offering very-

high-speed control of the direction of

light toward a digital display device.

Quick Quiz 38.7

If laser light is reflected from a phonograph record or a

compact disc, a diffraction pattern appears. This is due to the fact that both devices
contain parallel tracks of information that act as a reflection diffraction grating. Which
device, (a) record or (b) compact disc, results in diffraction maxima that are farther
apart in angle?

Quick Quiz 38.8

Ultraviolet light of wavelength 350 nm is incident on a dif-

fraction grating with slit spacing d and forms an interference pattern on a screen a
distance L away. The angular positions !

bright

of the interference maxima are large. The

locations of the bright fringes are marked on the screen. Now red light of wavelength
700 nm is used with a diffraction grating to form another diffraction pattern on the
screen. The bright fringes of this pattern will be located at the marks on the screen if

Silicon Light Machines

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