Transcript Physics 214 3: Interference Diffraction and Polarization

Physics 214
3: Interference Diffraction and Polarization
•Young’s 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
In order to observe interference
in light rays, light must be:
•Coherent
•Monochromatic
Superposition Principle must
apply
Double Slit Experiment
In phase
Out of phase
x axis
L
r1
q
d
r2
d
P; (x=0)
y
path difference
d = r 2 - r1
d
sin q =
d
 d = dsin q = r 2 - r 1
We get constructive interference when
d = dsin q = ml, m = 0, 1, 2,K
We get destructive interference when

1 
d = dsin q = m +  l

2 
for small
q
ml
y
q

q
=
= sin
tan
L
d
\ position of FRINGES
lL
y bright = m
d
lL
y dark = m + 1 2
d
Consider electric field intensity of
the two interfering light waves at the point P
(
)
E 1 = E 0 sin w t
E 2 = E 0 sin ( w t + f )
f only depends on path difference d
path difference of one wavelength l
c
phase difference of 2p radians
l
path difference of
2
c
phase difference of p radians
d
f
d l
\ =
 =
l 2p
f 2p
2p
2p
\f =
d=
dsin q
l
l
i.e. f = f(q)
Electric field intensity at point P, Ep
E p = E1 + E2 = E0 (sin wt + sin(wt + f))
f
= 2 E0 cos sin (wt + f)
2
1424
3
Amplitude
f = 0,2 p,K  constructive interference
f = p,3p,K  destructive interference
Intensity I of combined wave
I  E p2 max
Amplitude squared
Intensity of an electromagnetic wave is given by
2
2
EmaxBmax Emax
cBmax
I = Sav =
=
=
= cuav
2m 0
2m0 c 2m0
f
4E0 cos
2f
2 f
2
=
= 4I0 cos = Imax cos
2m 0 c
2
2
2
\ Itot
\ Itot
2
pdsin q
= Imax cos 

 l 
2
y
as sin q  we obtain
L


2 pd
Itot = Imax cos  y
lL 
Interference by Thin Films
white
light
1
2
1800 phase change
no phase change
air
soap
air
Get destructive and constructive
interference depending on wavelength
and position of observer: therefore see
colors at different positions.
If ray 1 is 180 0 out phase with ray 2 this
ln
is equivalent to a path difference of
2
 wavelength of light in medium



l 

l
whose refraction index is n and n = 


n 
ln
if 2 t =
rays will recombine in phase, in general
2


1 
1 
2t = m +  l n  2 nt = m +  l , m = 0,1, 2, 3, K


2 
2 
constructive interference
2 nt = ml, m = 0,1, 2, 3, K
destructive interference
Interference by Thin Films
1800 phase change
white
light
2nt
1800 phase change
air
oil
water
=
m l ,
m
= 0, 1 , 2, 3, K
constructive interference
2 nt
 m +

1 
l ,
m

2
destructive interference
=
= 0, 1 , 2, 3, K
t
Spreading out of
light is called
DIFFRACTION
This can occur
when light passes
through small
opening
or around object at
sharp edges
•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
P
a/2
a/2
q
Single Slit
In Fraunhofer Diffraction paths of waves are parallel
wave 1 travels further than wave 3 by amount
= path difference = d =
a
sin q
2
same for waves 2
l
( phase shift of p ) waves cancel through
2
destructive interference. This is true for any waves
If d =
a
. \ waves from upper half
2
destructively interfere with waves from bottom half
l
l
a
q
=

q
=
sin d
sin d
2
2
a
The argument holds when dividing slit into 4 portions
l
a
2l
q
=

q
=
sin d
sin d
4
2
a
l
 sin q d = m ; m =  1,K
a
that differ by
& 4.
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


pa
sin q
sin

l
 = E 0 sinc
E q = E 0 
pa

sin q 


l
{
}
{
pa
sin q
l
and thus the intensity of radiation by
pa
2
I q = I 0 sinc
sin q
l
l

minima occur at sin q = m ; m =  1, K
a
( as we found before)
{
}
}
Resolving between closely spaced
sources
diffraction
pattern for
two separate
source points
for
sources
closer
together
Sources so
close that
they cannot
be resolved
•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
sinq =
l
a
 q ( as
l < < a

q is small)
l
a
q subtended by 2 sources must be
 qm
so
qm
=
in order to be resolved
For circular apertures of diameter D
qm
= 1 . 22
l
D
Diffraction Grating
d
P
q
f
d
d=dsinq
d = slit spacing
If d =m l = d sinq, m = 0, 1,K
waves from all slits will be in phase at P
 bright line at P; m is order # of diffraction pattern
mth order max. for each l occurs at some specific q
All l’ s are seen at m =0 q =0
l
m = 1  sinql =
d
2l
m = 2  sinql =
d
Resolving power of
diffraction grating
l ave
l ave
=
= Resolving power
R=
l 2 - l1 Dl
l1 , l 2 two wavelengths that can be just resolved
l 1  l  l 2 ; l1  l 2
gratings with high resolving power can
distinguish small differences in l
R = Nm ; N = # of lines of grating
= resolving power of mth order diffraction
for m=0 all wavelengths are
indistinguishable
for m=2 for grating with N=5000
R=5000X2=10000
therefore min. line
separation for just resolving
for an average wavelength of
600 nm
is
6x10 -2 nm
Diffraction by Crystals
atomic spacings in crystals are approx.
10 -10 nm and therefore can act as 3D
diffraction grating
d
q
condition for constructive interference
2dsin q = ml, m =1,
 Braggs Law
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
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 .
•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
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 I = I0 cos2 q
Malus’s Law
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).
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.
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