phy203_5

1
Physics 214
3 Interference Diffraction and Polarization

• Youngs Double-Slit Experiment
• Intensity Distribution of the Double-Slit
  Interference Pattern
• Interference in Thin Films
• Single Slit Diffraction
• Diffraction Grating
• Diffraction by Crystals
• Polarization of Light Waves

2

• In order to observe interference in light rays,
  light must be
• Coherent
• Monochromatic
• Superposition Principle must apply

Double Slit Experiment
3
In phase
Out of phase
x axis
L
P (x0)
r1
y
q
d
r2
d
4
d
-
path difference

r
r
2
1
d
q

sin
d
Þ
d

q
-
dsin

r
r
2
1


We get constructive interference when
d
l



q
m
m

dsin

,

0
,
1
,
2
,
K
We get destructive interference when
æ
ö
1
d
q

l
ç

m

dsin

è
ø
2


5
q
for small
l
m
y
q

q


sin
tan
L
d
\
position of FRINGES
l
L

y
m
bright
d
l
L
(
)
1


y
m
2
dark
d
Consider electric field intensity of
the two interfering light waves at the point P

w
E
E
t
sin
1
0
(
)

w

f
E
E
sin
t
0
2
f
d
only depends on path difference
l
path difference of one wavelength
c
p
phase difference of 2
radians


6
l
path difference of
2
c
p
phase difference of
radians
d
f
d
l
\

Þ

l
p
f
p
2
2
p
p
2
2
\
f

d

q
dsin
l
l
(
)
f
f
q
i.e.



7
E
Electric field intensity at point P,

p
(
)
(
)



w

w

f
E
E
E
E
sin
t
sin
t
p
1
2
0
f
(
)

w

f
E
t
2
cos
sin
0
2
1
2
4
3
4
Amplitude
f

p
Û
0
,
2
,
constructive interference
K
f

p
p
Û
,
3
,
destructive interference
K
Intensity I of combined wave
µ
2
E
I


p max
Amplitude squared
8
Intensity of an electromagnetic wave is given by
E
B
E
2
B
2
c

S
cu
I

max
max

max

max


av
av
m
m
m
c
2
2
2
0
0
0
f
E
2
2
4
cos
f
f
0
2
\



I
I
2
I
2
4
cos
cos
m
0
max
tot
c
2
2
2
0
æ
ö
p
q
dsin
\

2
I
I
cos
ç

è
ø
tot
max
l
y
q

as sin
we obtain
L
æ
ö
p
d

I
I
2
y
ç

cos
è
ø
max
l
tot
L

9
Interference by Thin Films
1800 phase change
1
white light
no phase change
2
air
soap
air
Get destructive and constructive interference
depending on wavelength and position of
observer therefore see colors at different
positions.
10
0
If ray 1 is 180
out phase with ray 2 this
l
is equivalent to a path difference of
n

2
é
ù
wavelength of light in medium
ê
ú
l
ê
ú
l
whose refraction index is n and


ê
ú
ë
û
n
n
l

t
if
2
n
rays will recombine in phase,
in general
2
æ
ö
æ
ö
1
1

ç

l
Û


ç

l
t
m
nt
m
m
2

2
,


0,
1,
2,
3,
K
è
ø
è
ø
n
2
2
constructive interference

l
2
nt
m
,

m

0,
1,
2,
3,
K
destructive interference


11
Interference by Thin Films
1800 phase change
white light
1800 phase change
air
oil
t
water
2
nt

m
l
,

m

0,
1
,
2,
3,
K
constructive interference
1
æ
ö
2
nt

m

l
,

m

0,
1
,
2,
3,
K
è
ø
2
destructive interference


12
Spreading out of light is called DIFFRACTION
This can occur when light passes through small
opening or around object at sharp edges
13

• Fraunhofer Diffraction
• Light forms plane waves when reaching screen
• long distance from source
• by converging lens
• Fresnel Diffraction
• Wavefronts are not plane waves
• short distance from source

14
P
?
a/2
a/2
Single Slit
15
In Fraunhofer Diffraction paths of waves are
parallel
wave 1 travels further than wave 3 by amount
a

d
q

path difference

sin
same for waves 2

4.
2
l
(
)
d
Û
p
If

phase shift of
waves cancel through
2
destructive interference. This is true for any
waves
a
\
that differ by
.
waves from upper half
2
destructively interfere with waves from bottom
half
l
l
a
q

Û
q

sin
sin
d
d
2
2
a
The argument holds when dividing slit into 4
portions
l
l
a
2
q

Û
q

sin
sin
d
d
4
2
a
l
Þ
q



m
m
sin


1
,
K
d
a


16
By using the method of phasors one can
find that the electric field at a point P
on the screen due to radiation from all
points within the slit is given by


æ
ö
p
a


q
sin
sin
ç

p
a
l


q
ç

E
E
E
sin
c
sin
p
a
q
0
0
l
ç

q
sin
è
ø
l
and thus the intensity of radiation by


p
a

2
q
I
I
sin
c
sin
q
0
l
l
Þ
q


m
m
minima occur at sin



1
,
K
a
(
)
as we found before


17
Resolving between closely spaced sources
Sources so close that they cannot be resolved
diffraction pattern for two separate source
points
for sources closer together
18

• Rayleighs Criterion
• when central max. of one image falls on first
  min. of other image, the images are said to be
  just resolved

first min in single slit occurs when
l
sin
q


q

(
as
l
lt
lt
a
Þ
q
is small
)
a
l
so
q

m
a
q
subtended by
2
sources must be
³
q
m
in order to be resolved
For circular apertures of diameter D
l

q

1
.
22

m
D

19
Diffraction Grating
d
P
?
?
?
????dsin? d slit spacing
20
d
m
l

sin
q


d
m
If

,

0
,
1
,
K
waves from all slits will be in phase at P
Þ
m
bright line at P

is order
of diffraction pattern
l
th
m
order max. for each
occurs at some specific
q
Û
q
m
All

s are seen at

0

0
l
l

Þ
q

m
1
sin
l
d
l
2

Þ
q

m
2
sin
l
d


21
Resolving power of diffraction grating
l
l



R
ave
ave
Resolving power
l
-
l
D
l
2
1
l
l
,
two wavelengths that can be just resolved
1
2
l

l

l
l

l


1
2
1
2
gratings with high resolving power can
l
distinguish small differences in

R
Nm
N




of lines of grating
m
th

resolving power of
order diffraction

22
for m0 all wavelengths are indistinguishable for
m2 for grating with N5000 R5000X210000 theref
ore min. line separation for just resolving for
an average wavelength of 600 nm is 6x10 -2 nm
23
Diffraction by Crystals
atomic spacings in crystals are approx. 10 -10
nm and therefore can act as 3D diffraction
grating
d
?
24
Polarization
Electromagnetic Radiation is made of oscillating
electric and magnetic fields, that are
perpendicular to each other and to the direction
of propagation of the radiation (Transverse
Wave). These fields are proportional to each
other in magnitude and are in phase.
E
B
25
In general radiation is made up of a mixture of
such fields, with each wave of light having
different orientation i.e
as the electric vectors are always perpendicular
to the magnetic ones we need only show the
electric ones .
26

• Plane Polarized Light
• Electric Field is in only one direction.
• Light is Linearly Polarized
• E direction is constant in time
• Light is Circularly Polarized
• E rotates
• Ex Ey at all times
• Light is Elliptically Polarized
• E rotates
• Ex Ey at all times

27
Producing Polarization can produce such light by
passing through a polaroid sheet (Diochroic
Material) this allows only one orientation of
electric field through undiminished and
completely absorbs the light with electric fields
perpendicular to this direction. In general
diminishes the intensity according to
Maluss Law
28
polarized light is also produced by
reflection When light strikes a nonmetallic
surface at any angle other than perpendicular,
the reflected beam is polarized preferentially
in the plane parallel to the surface. (light
polarized in plane perpendicular to surface is
preferentially absorbed or transmitted).
29
Why is the Sky Blue and daylight polarized?
Polarization by Scattering

• Higher frequencies are scattered more than lower
  ones (refracted more) by the oxygen and nitrogen
  molecules
• All the visible frequencies are scattered the
  same by larger objects e.g. water droplets in
  clouds.
• Scattered light is polarized.

30
Polarization by Double Refraction

• Materials that have two indices of refraction
  depending on the direction of incident rays are
  called Double Refracting or Birefringent
• These materials produce polarized light