21 Chapters & 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 t + )
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 t + 1 )
r1
+ A(r2 ) cos( kr2 t + 2 )
r2
Sources will interfere constructively
when
( kr1 + 1 ) ( kr2 + 2 ) = 2 m
m = 0, 1, 2, ...
Sources will interfere destructively
when
1
( kr1 + 1 ) ( kr2 + 2 ) = 2 m +
2
Field and Intensity far from sources*
E (r, t ) = A(r1 ) cos( kr1 t ) + A(r2 ) cos( kr2 t )
Trigonometry
suppose
A(r1 ) = A(r2 )
E (r, t ) 2 A cos(
kr
) cos( kr t )
2
Field amplitude
depends on space
*Special case
r = r1 r2
r +r
r= 1 2
2
Field oscillates in
time.
1 = 2 = 0
cos( A) + cos( B)
A+ B
AB
= 2 cos(
) cos(
)
2
2
Average intensity depends in difference in distance to sources, r
Field
kr
E (r, t ) 2 A cos(
) cos( kr t )
2
r = r1 r2
r +r
r= 1 2
2
I ave
1.5
1
1 0
k r
=
2 A cos(
)
2 0
2
2
1.2
1
kr
)
2
0.8
Intensity
0.5
E
2 A cos(
I=
Intensity
0 ! 2
E
0
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 + 1 ) ( kr2 + 2 ) = kr = 2 m
Phases same because source comes
from a single incident plane wave
Dark fringes
kr = kd sin = 2 m
sin m m = m / d
m = 0, 1, 2, ...
1
sin m m = m + / d
2
Intensity on a distant screen
I ave
1 0
k r
=
2 A cos(
)
2 0
2
kr = kd sin kd =
Fringe spacing
I=
0
2
E
0
2
L
2 y
d
L
y =
L
d
Intensity from a single source
1 0
2
I1 =
A
2 0
Maximum Intensity at fringe
I =
1 fringe 0
2
2 A = 2 I1
2 0
d
Real pattern affected by slit opening width and distance to screen
Diffraction Grating
N slits, sharpens bright fringes
Bright fringes at same
angle as for double slit
sin m = m / d
m = 0, 1, 2, ...
Location of Fringes on distant screen
sin m = m / d
ym
= tan m
L
Intensity on a distant screen
Average over time
I ave
I=
0
2
E
0
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
sin m = m / d
ym
= tan m
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.
sin m = m / d
ym
= tan m
L
Accurate resolution of
spectrum requires many
lines
Reflection Grating
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 Maxwells
equations. (wave equation)
That is not always so easy.
In the past, not possible.
In the distant past equations
were not known.
Huygens (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.
C. Huygens
At a later time, the shape of the wave front is the
line tangent to all the wavelets.
R. Plant
Not the same
person.
Wikimedia Commons
www.guerrillacandy.com/.../
Huygens Principle:
When is there perfect destructive interference?
Destructive when
a
r12 = sin =
2
2
1 cancels 2
3 cancels 4
5 cancels 6
Etc.
Also:
a
sin p =
2p
2
p = 1, 2, 3.....