Download Fig. 36-6

Chapter 36
Diffraction
In Chapter 35, we saw how light beams passing through different slits can
interfere with each other and how a beam after passing through a single slit
flares—diffracts—in Young's experiment. Diffraction through a single slit or
past either a narrow obstacle or an edge produces rich interference patterns.
The physics of diffraction plays an important role in many scientific and
engineering fields.
In this chapter we explain diffraction using the wave nature of light and
discuss several applications of diffraction in science and technology.
(36-1)
36.2 Diffraction and the Wave Theory of Light
Diffraction pattern from a single narrow slit.
Side or secondary
maxima
Light
Central
maximum
These patterns cannot be
explained using geometrical
optics (Ch. 34)!
Fresnel Bright Spot.
Light
Bright
spot
(36-2)
36.3 Diffraction by a Single Slit: Locating the Minima
When the path length difference between rays r1
and r2 is l/2, the two rays will be out of phase when
they reach P1 on the screen, resulting in destructive
interference at P1. The path length difference is the
distance from the starting point of r2 at the center of
the slit to point b.
For D>>a, the path length difference between rays
r1 and r2 is (a/2) sin q.
Fig. 36-4
(36-3)
Diffraction by a Single Slit: Locating the Minima, cont'd
Repeat previous analysis for pairs of rays, each separated by a
vertical distance of a/2 at the slit.
Setting path length difference to l/2 for each pair of rays, we
obtain the first dark fringes at:
a
l
sin q   a sin q  l
2
2
(first minimum)
For second minimum, divide slit into 4 zones of equal widths
a/4 (separation between pairs of rays). Destructive interference
occurs when the path length difference for each pair is l/2.
a
l
sin q   a sin q  2l (second minimum)
4
2
Dividing the slit into increasingly larger even numbers of zones,
we can find higher order minima:
a sin q  ml , for m  1, 2,3
Fig. 36-5
(minima-dark fringes)
(36-4)
36.4 Intensity in Single-Slit Diffraction, Qualitatively
To obtain the locations of the minima, the slit was equally divided into N zones,
each with width Dx. Each zone acts as a source of Huygens wavelets. Now
these zones can be superimposed at the screen to obtain the intensity as a
function of q, the angle to the central axis.
To find the net electric field Eq (intensity a Eq2) at point P on the screen, we
need the phase relationships among the wavelets arriving from different zones:
 phase   2


 difference   l
  path length 
  difference 


 2 
D  
  Dx sin q 
 l 
N=18
1st side
max.
q=0
1st min.
q small
Fig. 36-6
(36-5)
36.5 Intensity in Single-Slit Diffraction,
Quantitatively
Here we will show that the intensity at the screen due to
a single slit is:
 sin a 
I q   I m 
(36-5)

 a 
1
a
where a   
sin q (36-6)
2
l
2
In Eq. 36-5, minima occur when:
a  m ,
for m  1, 2,3
If we put this into Eq. 36-6 we find:
Fig. 36-7
a
m 
sin q , for m  1, 2,3
l
or a sin q  ml , for m  1, 2,3
(minima-dark fringes)
(36-6)
Proof of Eqs. 36-5 and 36-6
If we divide the slit into infinitesimally wide zones Dx, the arc of the phasors
approaches the arc of a circle. The length of the arc is Em.  is the difference in
phase between the infinitesimal vectors at the left and right ends of the arc.  is
also the angle between the 2 radii marked R.
Eq
.
The dashed line bisecting f forms two triangles, where: sin  
2R
Em
.
In radian measure:  
R
Em
1
sin
Solving the previous 2 equations for Eq one obtains: Eq 
2 .
1
1
2
2
The intensity at the screen is therefore:
I q  Eq2
 sin a 
 2  I q   I m 

Im
Em
a



2
 is related to the path length difference across the
entire slit:
Fig. 36-8
 2 
     a sin q 
 l 
(36-7)
36.6 Diffraction by a Circular Aperture
Distant point
source, e,g., star
d
lens
sin q  1.22
l
d
q
(1st min.- circ. aperture)
Image is not a point, as
expected from geometrical
optics! Diffraction is
responsible for this image
pattern.
a
Light
Light
sin q 
q
l
a
(1st min.- single slit)
a
q
(36-8)
Resolvability
Rayleigh’s Criterion: Two point sources are barely resolvable if their angular
separation qR results in the central maximum of the diffraction pattern of one
source’s image centered on the first minimum of the diffraction pattern of the
other source’s image.
Fig. 36-10
q R small
l
l


1
q R  sin 1.22   1.22
d
d

(Rayleigh's criterion)
(36-9)
36.7
Diffraction by a Double Slit
In the double-slit experiment described in Ch. 35 we assumed that the slit width
a<<l. What if this is not the case?
Two vanishingly narrow slits a<<l
Single slit a~l
Fig. 36-14
Two Single slits a~l
 sin a 
I q   I m  cos   

a


2
2
(double slit)
d

sin q
l
a
a
sin q
l
(36-10)
36.8 Diffraction Gratings
A device with N slits (rulings) can be used to manipulate light, such as separate
different wavelengths of light that are contained in a single beam. How does a
diffraction grating affect monochromatic light?
Fig. 36-17
d sin q  ml for m  0,1, 2
Fig. 36-18
(maxima-lines)
(36-11)
Width of Lines
The ability of the diffraction grating to resolve (separate) different wavelengths
depends on the width of the lines (maxima).
Fig. 36-20
Fig. 36-19
(36-12)
Width of Lines, cont’d
Nd sin Dqhw  l ,
Dq hw 
Fig. 36-21
Dq hw 
l
Nd cosq
l
Nd
sin Dqhw  Dqhw
(half width of central line)
(half width of line at q )
(36-13)
Grating Spectroscope
Separates different wavelengths (colors) of light into distinct diffraction lines
Fig. 36-23
Fig. 36-22
(36-14)
Optically Variable Graphics
Gratings embedded in the device send out hundreds or even thousands of
diffraction orders to produce virtual images that vary with the viewing angle.
This is complicated to design and extremely difficult to counterfeit, so it makes
an excellent security graphic.
Fig. 36-25
(36-15)
36.9 Gratings: Dispersion and Resolving Power
Dispersion: the angular spreading of different wavelengths by a grating
Dq
D
(dispersion defined)
Dl
m
D
(dispersion of a grating) (36-30)
d cos q
Resolving Power
lavg
R
Dl
(resolving power defined)
R  Nm (resolving power of a grating) (36-32)
(36-16)
Proof of Eq. 36-30
Angular position of maxima
Differential of first equation (what
change in angle does a change in
wavelength produce?)
For small angles
d sin q  ml
d  cosq  dq  md l
dq  Dq and dl  Dl
d  cos q  Dq  mDl
Dq
m

Dl d  cos q 
(36-17)
Proof of Eq. 36-32
l
Rayleigh's criterion for half width
to resolve two lines
Dq hw 
Substituting for Dq in calculation on
previous slide
Dq hw  Dq

l
N
Nd cos q
 mDl
l
R
 Nm
Dl
(36-18)
Dispersion and Resolving Power Compared
Table 36-1
Grating N
d (nm)
q
D (o/mm)
R
A
10 000 2540
13.4o
23.2
10 000
B
20 000 2540
13.4o
23.2
20 000
C
10 000 1360
25.5o
46.3
10 000
Data are for l = 589 nm and m = 1
Fig. 36-26
(36-19)
36.10 X-Ray Diffraction
X-rays are electromagnetic radiation with wavelength ~1 Å = 10-10 m (visible
light ~5.5x10-7 m).
X-ray generation
X-ray wavelengths too short to be resolved
by a standard optical grating
Fig. 36-27
ml
1 1 0.1 nm 
q  sin
 sin
 0.0019
d
3000 nm
1
(36-20)
X-Ray Diffraction, cont’d
Diffraction of x-rays by crystal: spacing d of
adjacent crystal planes on the order of 0.1 nm
→ three-dimensional diffraction grating with
diffraction maxima along angles where reflections
from different planes interfere constructively
2d sin q  ml for m  0,1, 2
Fig. 36-28
(Bragg's law)
(36-21)
X-Ray Diffraction, cont’d
Interplanar spacing d is related to the
unit cell dimension a0:
5d 
Fig. 36-29
5
4
2
0
a
a0
or d 
 0.2236a0
20
Not only can crystals be used to
separate different x-ray wavelengths,
but x-rays in turn can be used to study
crystals, for example, to determine the
type of crystal ordering and a0.
(36-22)