1. 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 1

2. Double Slit Experiment
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; (x=0) r1 y q d r2 d

4. We get constructive interference when
path difference d = r - r 2 1 d
sin = q d Þ d =
dsin q = r - r 2 1
We get constructive interference when d =
dsin q = m l , m = , 1 , 2 , K
We get destructive interference when æ 1 ö d =
dsin q = ç m + ÷ l è ø 2

5. ( ) (kx-wt) (kx-wt+f) for small q m l y = sin q » tan q = d L \
position of FRINGES l L y = m
bright d ( ) l L y = m + 1 2
dark d
Consider electric field intensity of
the two interfering light waves at the point P
(kx-wt) E = E
sin 1
(kx-wt+f) E = E
sin 2 f
only depends on path difference d
path difference of one wavelength l c
phase difference of 2 p
radians

6. ( ) l path difference of 2 c phase difference of p radians d f d l \ =Þ = l 2 p f 2 p 2 p 2 p \ f = d =
dsin q l l
i.e. f = f ( q )

7. ( ) Electric field magnitude at point P, E E = E + E = E f = 2 E cos
(kx-wt)
(kx-wt+f)
sin +
sin p 1 2 f = 2 E
cos
sin
(kx-wt+f/2) 2 1 4 2 4 3
Amplitude f = , 2 p , Û K
constructive interference f = p , 3 p , Û K
destructive interference
Intensity I of combined wave I E 2
p max
Amplitude squared

8. E B E c B I = S = = = = cu 2 m 2 m c 2 m f 4 E cos f f 2 \ I = = 4 I
Intensity of an electromagnetic wave is given by E B E 2 c B 2 I = S =
max
max =
max =
max = cu av 2 m 2 m c 2 m av f 4 E 2
cos 2 f f 2 \ I = = 4 I
cos 2 = I
cos 2
tot 2 m c 2
max 2 æ p q ö
dsin \ I = I
cos 2 ç ÷
tot
max è l ø y
as sin q
we obtain L æ p ö d I = I
cos 2 ç y ÷
tot
max è l L ø

9. Interference by Thin Films1
1800 phase change
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. is equivalent to a path difference of 2 é ù
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 ê l ú
whose refraction index is n is l = ê ë ú n û n l if 2 t = n
rays will recombine in phase,
in general 2 æ 1 ö æ 1 ö 2 t = ç m + ÷ l Û 2 nt = ç m + ÷ l , m = 0, 1, 2, 3, K è ø è ø 2 n 2
constructive interference 2 nt = m l , m = 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 = 1 , 2, 3, K
constructive interference æ 1 ö 2 nt = m + l , m = 0, 1 , 2, 3, K è 2 ø
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 =
path difference = d =
sin q
same for waves 2
& 4. 2 l ( ) If d = Û
phase shift of p
waves cancel through 2
destructive interference. This is true for any waves a
that differ by . \
waves from upper half 2
that destructively interfere with waves from bottom half are at angle qd a l l
sin q = Û
sin q = 2 d 2 d a
The argument holds when dividing slit into 4 portions a l 2 l
sin q = Û
sin q = 4 d 2 d a l Þ
sin q = m ; m = 1 , K d a

16. { } { } { } Sinc 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 p a ö ç
sin
sin q { } ÷ p a l E = E ç ÷ = E
sin c
sin q p q a l ç
sin q ÷ è l ø
and thus the intensity of radiation by { } p a I = I
sin c 2
sin q q l l Þ
minima occur at sin q = m ; m = 1 , K a
Sinc

18. Fresnel / Fraunhofer Diffraction from a Single Slit
Far from the slit z
Close to the slit
Slit
Incident plane wave

19. Resolving between closely spaced sources
diffraction
pattern for
two separate
source points that
can
be resolved
sources closer together
that can be just
resolved
Sources so close that they cannot be resolved

21. 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
<
< a Þ q
is small ) a l so q =
min a q
subtended by 2
sources must be q
min
in order to be resolved
For circular apertures of diameter D l q = 1 . 22
min D

22. Diffraction Grating d P   
dsin
d = slit spacing

23. waves from all slits will be in phase at PIf d = m l = d
sin q , m = , 1 , K
waves from all slits will be in phase at P Þ
bright line at P; m
is order #
of diffraction pattern m th
order max. for each l
occurs at some specific q
All l ’
s are seen at m = Û q = l m = Þ
sin 1 q = l d 2 l m = 2 Þ
sin q = l d

24. Resolving power of diffraction grating=
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
distinguish small differences in l R = Nm ; N = #
of lines of grating =
resolving power of m th
order diffraction

25. 600 nm is 6x10 -2 nm for m=0 all wavelengths are indistinguishable
for m=2 for grating with N=5000
R=5000X2=10000
therefore min. wavelength difference that can be resolved for waves with an average wavelength of
600 nm is
6x10 -2 nm

26. Diffraction by Crystals atomic spacings in crystals are approx.
nm and therefore can act as 3D diffraction grating d 

27. 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

28. 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 .

29. 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
Ex Ey at all times

30. 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
Malus’s Law

31. 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).

32. 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.

33. 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