Chapters 21 & 22
Interference and Wave Optics
Waves that are coherent can add/cancel
Patterns of strong and weak intensity
Single Spherical Source
Approximate Electric Field:
E(r,t) = A(r)cos(kr - wt + q)
Field depends on distance from source
and time.
Typically
A(r) : 1 / r
Most important dependence is in the
cosine
Two sources that have exactly the
same frequency. “Coherent”
E(r,t) = A(r1 )cos(kr1 - wt + f 1 )
r1
r2
+ A(r2 )cos(kr2 - wt + f 2 )
Sources will interfere constructively
when
(kr1 + f 1 ) - (kr2 + f 2 ) = 2pm
m = 0, 1, 2, ...
Sources will interfere destructively
when
Ê
m+
( kr1 + f 1 ) - ( kr2 + f 2 ) = 2p Á
Á
Á
Ë
1 ˆ˜
˜
2 ˜¯
Incoherent vs Out of Phase
Incoherent
Two signals have different
frequencies. Sometimes the same
sign, sometimes opposite signs.
1.5
1.5
1
1
0.5
0.5
Field
Field
Coherent, but out of phase.
Two signals have the same
frequency, but one leads or lags the
other.
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-5
0
5
10
15
t
20
25
30
35
-5
0
5
10
15
t
20
25
30
35
Field and Intensity far from sources*
E(r,t) = A(r1 )cos(kr1 - wt) + A(r2 )cos(kr2 - wt)
Trigonometry
suppose
A(r1 ) = A(r2 )
E(r,t) ; 2A cos(
kDr
)cos(kr - wt)
2
Field amplitude
depends on space
*Special case
cos(A) + cos(B)
A+ B
A- B
= 2 cos(
)cos(
)
2
2
Dr = r1 - r2
r+r
r= 1 2
2
Field oscillates in
time.
f1= f2= 0
Average intensity depends in difference in distance to sources, r
Field
Dr = r1 - r2
r+r
r= 1 2
2
I ave
kDr
)
2
2
1
1
0.8
Intensity
0.5
E
1 0
kr

2A cos(
)
2 0
2
1.2
1.5
2A cos(
0 r 2
I
E
0
Intensity
kDr
E(r,t) ; 2A cos(
)cos(kr - wt)
2
0
0.6
0.4
-0.5
0.2
-1
0
-1.5
-15
-10
-5
0
t
5
10
15
-15
-10
-5
0
t
5
10
15
Interference of light
Coherence because sources are
at exactly the same frequency
Sources will interfere constructively when
(kr1 + f 1 ) - (kr2 + f 2 ) = kDr = 2pm
Phases same because source comes
from a single incident plane wave
Dark fringes
kDr = kd sinq = 2pm
sinqm ª qm = ml / d
m = 0, 1, 2, ...
Ê
1 ˆ˜
sinqm ª qm = Á
m
+
˜˜ l / d
Á
Á
Ë
2¯
Intensity on a distant screen
I ave
1 0
kr

2A cos(
)
2 0
2
kDr = kd sin q ; kd q =
Fringe spacing
Dy =
I=
e0 2
E
m0
2
L
2p y
d
l
L
Ll
d
Intensity from a single source
1 0
2
I1 
A
2 0
Maximum Intensity at fringe
I fringe
1 0
2

2A  2I1
2 0
d
Real pattern affected by slit opening width and distance to screen
Suppose the viewing screen in
the figure is moved closer to
the double slit. What happens
to the interference fringes?
A. They fade out and disappear.
B. They get out of focus.
C. They get brighter and closer together.
D. They get brighter and farther apart.
E. They get brighter but otherwise do not change.
Light of wavelength l1 illuminates a
double slit, and interference fringes are
observed on a screen behind the slits.
When the wavelength is changed to l2,
the fringes get closer together. How large
is l2 relative to l1?
A. l2 is smaller than l1.
B. l2 is larger than l1.
C. Cannot be determined from this information.
Diffraction Grating
N slits, sharpens bright fringes
Bright fringes at same
angle as for double slit
sinqm = ml / d
m = 0, 1, 2, ...
Location of Fringes on distant screen
sinqm = ml / d
ym
= tan qm
L
Intensity on a distant screen
I ave 
Average over time
I=
e0 2
E
m0
1
I
2
Intensity from a single slit
amplitude from a single slit
1 0
2
I1 
A
2 0
At the bright fringe N slits interfere constructively
I fringe 
1 0
2
NA  N 2 I1
2 0
Spatial average of intensity must correspond to sum of N slits
I SA  NI1
I fringe
I SA
N
width of fringe
fringe width
1

fringe spacing N
sinqm = ml / d
ym
= tan qm
L
Measuring Light Spectra
Light usually contains a superposition of many frequencies.
The amount of each frequency is called its spectrum.
Knowing the components of the spectrum tells us about
the source of light.
Composition of stars is known by measuring the spectrum of their
light.
sinqm = ml / d
ym
= tan qm
L
Accurate resolution of
spectrum requires many
lines
White light passes through a diffraction
grating and forms rainbow patterns on a
screen behind the grating. For each
rainbow,
A.the red side is farthest from the center of the
screen, the violet side is closest to the center.
B.the red side is closest to the center of the screen,
the violet side is farthest from the center.
C.the red side is on the left, the violet side on
the right.
D.the red side is on the right, the violet side on
the left.
Reflection Grating
Incoherent vs Out of Phase
Incoherent
Two signals have different
frequencies. Sometimes the same
sign, sometimes opposite signs.
1.5
1.5
1
1
0.5
0.5
Field
Field
Coherent, but out of phase.
Two signals have the same
frequency, but one leads or lags the
other.
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-5
0
5
10
15
t
20
25
30
35
-5
0
5
10
15
t
20
25
30
35
Fields in slits are coherent
but out of phase
Diffraction pattern shifts
Field and Intensity far from sources
E(r,t) = A(r1 )cos(kr1 - wt + f 1 ) + A(r2 )cos(kr2 - wt + f 2 )
Trigonometry
suppose
E(r,t) ; 2A cos(
A(r1 ) = A(r2 )
kDr f 1 - f 2
f +f2
+
)cos(kr - wt + 1
)
2
2
2
Dr = r1 - r2
r=
Field amplitude
depends on space
r1 + r2
2
Field oscillates in
time.
Constructive interference when
cos(A) + cos(B)
A+ B
A- B
= 2 cos(
)cos(
)
2
2
D r = d sin q
kDr f 1 - f 2
+
= mp
2
2
Ê
f 1 - f 2 ˆ˜
d sinq = l Á
m
˜
Á
Á
Ë
2p ˜¯
Propagation of wave fronts from a slit with a nonzero width
Sources are not points.
How do we describe
spreading of waves?
Ans. Just solve Maxwell’s
equations. (wave equation)
That is not always so easy.
In the past, not possible.
In the distant past equations
were not known.
Huygen’s Principle
1.
Each point on a wave front is the source of a
spherical wavelet that spreads out at the wave
speed.
2.
At a later time, the shape of the wave front is the
line tangent to all the wavelets.
Huygen’s (1629-1695) Principle
1.
Each point on a wave front is the source of a
spherical wavelet that spreads out at the wave
speed.
2.
At a later time, the shape of the wave front is the
line tangent to all the wavelets.
C. Huygens
R. Plant
Not the same
person.
Wikimedia Commons
www.guerrillacandy.com/.../
Huygens Principle:
When is there perfect destructive interference?
Destructive when
Dr12 =
a
l
sin q =
2
2
1 cancels 2
3 cancels 4
5 cancels 6
Etc.
Also:
a
l
sin qp =
2p
2
p = 1,2, 3.....
We can calculate the pattern from a single slit!
Field from two point sources
E(r,t) = A(r1 )cos(kr1 - wt) + A(r2 )cos(kr2 - wt)
r1
r2
Field from many point sources
E(r,t) =
 A(r )cos(kr i
i
i
wt)
Field from a continuous distribution of point
sources - Integrate!
E(r,t) =
 A(r )cos(kr i
i
wt)
i
yi
dyi
a /2
E(r,t) =
dyi
Ú a A(ri )cos(kri - wt)
- a /2
Replace sum by integral
Distance from source to observation point
y=observation
point
ri
yi=source
point
y=0
a /2
E(r,t) =
L
ri =
dyi
Ú a A cos(kri - wt)
- a /2
L2 + (y - yi )2
Still can’t do integral. Must make an approximation,
ri ;
L2 + y2 -
yyi
L2 + y2
yi < < L, y
Result
r=
L2 + y 2
a /2
E(r,t) =
sin (Y)
dyi
A
cos(kr
wt)
;
A
cos(kr - wt)
i
Úa
Y
- a /2
Y=
Time average intensity
sin (Y)
1 e0
I ave =
A
2 m0
Y
kay
2r
1.2
2
1
0.8
0.6
Intensity zero when
Y = pp
0.4
0.2
0
p = ± 1, 2, 3,...
sin q =
y
pl
=
r
a
-0.2
-12
-8
-4
0

4
8
12
Fraunhofer Approximation
Named in honor of Fraunhofer
Fraunhofer lines
Absorption lines in sunlight
Joseph von Faunhofer
Wikimedia Commons
Width of Central Maximum
w 2l
=
r
a
What increases w?
1. Increase distance from slit.
2. Increase wavelength
3. Decrease size of slit
Circular aperture diffraction
Width of central maximum
w 2.44l
=
L
D
The figure shows two single-slit diffraction
patterns. The distance between the slit and the
viewing screen is the same in both cases. Which
of the following could be true?
A. The wavelengths are the same for both; a1 > a2.
B. The wavelengths are the same for both; a2 > a1.
C. The slits and the wavelengths are the same for both; p1 > p2.
D. The slits and the wavelengths are the same for both; p2 > p1.
Wave Picture vs Ray Picture
If D >> w, ray picture is OK
If D <= w, wave picture is needed
Critical size:
Dc = w
Dc =
w 2.44l
=
L
D
2.44l L
If product of wave length and distance to big, wave picture necessary.
Distant object
When will you see
D > Dc =
?
2.44l L
D
When will you see
Diffraction blurs image
Example suppose object
is on surface of sun
L = 1.5¥ 1011 m
l = 500nm = 5¥ 10- 7 m
Dc =
2.44l L = 427m
?
D £ Dc =
2.44l L
Interferometer
Sources will interfere constructively
when
Dr = 2L = ml
m = 0, 1, 2, ...
Sources will interfere destructively
when
Ê
1 ˆ˜
Dr = 2L = Á
m
+
˜˜ l
Á
Á
Ë
2¯
Dm
If I vary L
Dm =
DL
l /2
DL
What is seen
Michelson Interferometer
As L2 is varied, central spot
changes from dark to light,
etc. Count changes = m
If I vary L2
Dm =
DL2
l /2
Using the interferometer
Michelson and Morley showed
that the speed of light is
independent of the motion of
the earth.
This implies that light is not
supported by a medium, but
propagates in vacuum.
Led to development of the
special theory of relativity.
Albert Michelson
First US Nobel Science Prize Winner
Wikimedia Commons
Albert Michelson was the first US Nobel Science Prize
Winner. The first US Nobel Prize winner was awarded the
Peace Prize.
This American is known for saying:
A.
B.
C.
D.
Peace is at hand.
All we are saying, is give peace a chance.
There will be peace in the valley.
Speak softly, and carry a big stick.
A Michelson interferometer using light of wavelength l
has been adjusted to produce a bright spot at the center
of the interference pattern. Mirror M1 is then moved
distance l toward the beam splitter while M2 is moved
distance l away from the beam splitter. How many
bright-dark-bright fringe shifts are seen?
A. 4
B. 3
C. 2
D. 1
E. 0
Measuring Index of refraction
Number of
wavelengths in cell
when empty
m1 =
2d
l vac
Number of
wavelengths in cell
when full
m2 =
Number of fringe
shifts as cell fills up
2d
2d
=
l gas l vac / ngas
2d
Dm = m2 - m1 = (ngas - 1)
l vac
EXAMPLE 22.9 Measuring the
index of refraction
QUESTION:
EXAMPLE 22.9 Measuring the
index of refraction
What do we know?
2d
Dm = m2 - m1 = (ngas - 1)
l vac
EXAMPLE 22.9 Measuring the
index of refraction
2d
Dm = m2 - m1 = (ngas - 1)
l vac
EXAMPLE 22.9 Measuring the
index of refraction
Mach-Zehnder Interferometer
Adjustable delay
Interference depends on index of
refraction of unknown
Viewing screen or camera
source
Unknown material
displacement of interference fringes
gives “line averaged” product
(n - 1)
d
2d
l vac
Imaging a profile of index
change
Chapter 22. Summary Slides
General Principles
General Principles
Important Concepts
Applications
Applications
Applications
Applications