This document provides a summary of key concepts related to interference and diffraction of light. It discusses Huygen's principle, how light diffracts through single and double slits, and how this results in interference patterns of light and dark bands. It also covers how diffraction gratings can be used to separate light into different colors based on wavelength. The document derives mathematical expressions to describe these interference and diffraction phenomena. Key findings include how the spacing and number of slits or lines on a diffraction grating determine the angular separation of bands in the interference pattern.

1. SHIKHA CHOUHAN
BSC (C.S.) 3RD SEM
PHYSICS
INTERFERENCE OF LIGHT

2. Interference and Diffraction
Huygen’s Principle
•Any wave (including electromagnetic waves) is able
to propagate because the wave here affects nearby
points there
•In a sense, the wave is the source for more of
the wave
•A wave here creates waves in all the forward
directions
•For a plane wave, the generated waves add up to
make more plane waves •Mathematically, this
works, but for plane
waves, no one does it
this way

3. Diffraction Through a Tiny Hole
•The waves come out in all directions
•It is only because the whole wave makes
new waves that the waves add up to only go
forwards
•What if we let the wave pass through a tiny
hole?
•Smaller than a wavelength
•Only one point acts as
source
•Waves spread out in all
directions
 0 sinE E kx t 
 0
sin
E
E kr t
r


 r
 sinE kr t:
•What’s interesting is
that oscillations depend
on distance from slit

4. Interference Through Two Slits
•Now imagine we have two slits, equally sized
•Each slit creates its own waves
•In some directions,
crests add with crests to
make bigger “brighter”
crests
•In others, crests
combine with troughs
to make minimum areas
•In the end, what you
get is a pattern of
alternating light and
dark bands
•We’re about to
need an obscure
math identity:
sin sin 2sin cos
2 2
A B A B
A B
    
     
   

5. Interference Through Two Slits
(2)•What do the EM waves look like far away?
•Let the separation of the slits be d
•Let’s find total E-field at point P
d
P

d
sin
r
1
r2
1 2E E E     1 2~ sin sinkr t kr t   
   1
ave 2 122sin coskr t k r r    
 1
ave 1 22r r r 
2 1 sinr r d  
   1
ave 22sin cos sinE kr t kd :
   2 2 2 1
ave 2sin cos sinI E kr t kd : :

6. Interference Through Two Slits
(3)
 2 1
2cos sinI kd 
2
k


 2
max
sin
cos
d
I I
 

 
   
•Where is it
bright?
•Where is it
dark?
brightsin m d  0, 1, 2,m    K
 1
dark 2sin m d  

7. Interference Through Four Slits
•What if we have more than two
slits?
•Four slits, each spaced
distance d apart
•Treat it as two double slits
P
r1,
2
r3,
4
1 2 3 4E E E E E    1,2 3,4E E 
       1 1
1,2 3,42 22sin cos sin 2sin cos sinkr t kd kr t kd      
     ave4sin cos sin cos 2 sinkr t d d       
2
I E
•For four slits, every third
band is bright

8. More Slits and Diffraction
Gratings•This process can be continued
for more slits
•For N slits, every N – 1’th
band is bright
•For large N, bands become
very narrow
N =
8
N =
16
N =
32
•A device called a
diffraction grating is
just transparent with
closely spaced regular
lines on it
•You already used
it in lab
brightsin
m
d

  0, 1, 2,m    K
•Diffraction gratings are another way to divide
light into different colors
•More accurate way of measuring wavelength
than a prism

9. Resolution of Diffraction
Gratings•Note that the angle depends on the
wavelength
•With a finite number of slits, nearby
wavelengths may overlap •The width of the
peaks is about
•The difference
between peaks is
•We can distinguish
two peaks if:
1
1
2
2
sin
sin
m
d
m
d






 sin
dN

 
N =
8

1.1
 
 sin
m
d



 
•This quantity (mN) is called
the
resolving power
•Even if N is very large,
 m N
d d
 
 mN





10. Diffraction Through a Single Slit
•What if our slit is NOT small compared to a wavelength
•Treat it as a large number of closely spaced sources, b
Huygen’s principle
ra
ve
P
r
a x
•Let the slit size be a, and rave the
distance to the center
•Let x be the distance of some point
from the center
•The distance r will be slightly
different from here to P
ave sinr r x  
 ~ sinE kr t  avesin sinkr kx t   
 
1
2
1
2
avesin sin
a
a
E kr kx t dx 

  
 
/2
ave /2
1
cos sin
sin
a
a
kr kx t
k
 
 
   
 
 
1
ave 2
1
ave 2
cos sin1
sin cos sin
kr ka t
E
k kr ka t
 
  
  
   
    
L
 
 
1
2
ave1
2
sin sin
sin
sin
ka
kr t
k



 

11. Diffraction Through a Single Slit
(2)
 
2
1
2
max 1
2
sin sin
sin
ka
I I
ka


 
  
 
 
2
max
sin sin
sin
a
I I
a
  
  
 
  
 
darksin m a  1, 2,m    K
•Very similar to equation
for multi-slit diffraction,
but . . .
•a is the size of the slit
•This equation is for
dark, not light
•Note m= 0 is missing
•Central peak twice as
wide

12. Screens and Small Angles
•Usually your slit size/separation is large
compared to the wavelength
•Multi-slit: Diffraction:
•When you project them onto a screen, you need
to calculate locations of these bright/dark lines
•For small angles, sin and tan are the same
brightsin
m
d

  darksin
m
a

 
L
x
tan
x
L
 sin
bright
L
x m
d


dark
L
x m
a



13. Diffraction and Interference
Together
d
a
a
•Now go through two finite
sized slits
•Result is simply sum of each
slit
•Resulting amplitude looks like: 
   
1
2
1 21
2
sin sin
sin sin
sin
ka
E kr t kr t
k

 

    :
 
2
2
max
sin sin sin
sin
sin
a d
I I
a
    
   
   
    
  
a =
d/5
•Resulting pattern has two
kinds of variations:
•Fast fluctuations from
separation d
•Slow fluctuations from slit size

14. The Diffraction Limit
•When light goes through a “small” slit, its
direction gets changed
•Can’t determine direction better than this
a
a
•If we put light through rectangular
(square) hole,
we get diffraction in both
dimensions
•A circular hole of diameter D is a
trifle smaller, which causes a bit
more spread in the outgoing wave
•For homework, use this formula;
for tests, the approximate formula is
 minsin
a

 :
min
a

 :
D
min
1.22
D

  min
D

 :

15. Sample problem
•A degree is 1/360 of a circle, an
arc-minute is 1/60 of a degree, an
arc-second is 1/60 of an arc minute
•Telescopes require large apertures
to see small angles
min
1.22
D

 
If the pupil of your eye in good light is 2 mm
in diameter, what’s the smallest angle you
can see using 500 nm visible light?
7
3
1.22 5 10 m
2 10 m


 


4
3.05 10 rad
  1arc-min

16. Phases
•When you combine two (or more) waves, you need to
know the phase shift between them:
•The angle is the phase shift
•When the phase shift is zero, the waves add
constructively
•The result is bigger
•Same thing for any even multiple of 
•When the phase shift is , the waves add destructively
•The result is smaller
•Same thing for any odd multiple of 
•To find maximum/minimum effects, set phase shift to
even/odd multiples of 
   sinsinA x BE x   