physics III

1. Interference of light

2. 1. Two-slit interference
 Optical path L = nr (n: index of refraction, r: distance between two points)
 Notes
 d < 1 mm and i ~ sereval mm.
 Sign () implies the symmetry of interference fringes at two sides of central line.
 Interference max: L2 – L1 = dsin = m  position of bright fringe:
d
R
mλ
yb
m 
 Interference min:  position of dark fringe:
 
2
1
2
1
2
λ
m
θ
sin
d
L
L 


  
d
R
λ
m
yd
m
2
1
2 

 Width of fringe:
d
R
λ
y
y
i m
m 

 1
 Most important equations
S1
S2
P
(normally, is considered very small)

R
r1 = L1
r2 = L2
 = L2 – L1 = dsin
d
y = Rtg
H
O
r
Notes: d = distance; R = Range, L = optical path length
S1
S2
S
y, Intensity
Observation
screen
m = 2, 2nd order max.
m = 0, central max.
m = 1, 1st order max.
m = 2, 2nd order max.
m = 1, 1st order max.
m = 0, 1nd min.
m = 1, 2nd min.
m = 0, 1nd min.
m = 1, 2nd min.


m = 0, 1, 2, 3, …
 Model

3. 3
;
cos
d
BC
AB
2



where:
2
2
2
2
2
2
2
2
2
2
2
1
2 cos
2
cos
sin
1
2
sin
2
cos
2
θ
d
n
θ
θ
-
d
n
θ
dtgθ
n
θ
d
n
L
L 













AD = AC.sin1 = 2dtg2 sin1 và n1sin1 = n2sin 2
 Path difference between ABCE & ADF: L2 – L1 = n2(AB+BC) - n1AD
A
B
C
P
(1)
Two in-phase reflected
waves create
interference max.
Reflected ray on the
upper surface of film
1
2

D
n2
n1
(2)
d
Bright fringe
F
E
Reflected ray on the
lower surface of film
 Formation of interference fringes
2. Thin film interference

4. 4
2. Thin film interference
 
n
m
tb
m
4
1
2



 Bright fringe: (one phase shift ray) or (both phase shift rays)
 
n
m
td
m
4
1
2



 Dark fringe: (one phase shift ray) or both phase shift rays)
n
m
td
m
2


n
m
tb
m
2


 Film thickness should be < 1 m (corresponding to light’s wavelength)
 Observed position corresponding to the film thickness (t)
 Phase shift at the interface of two media, index of refraction n1 & n2
 if n1 > n2  no phase shift;
 if n1 < n2  phase shift of reflected ray;
1
2
1
2
2 t
Incidence
n2
A C
B
D
n1
2
Phase
shift
No phase shift
 Constructive condition:  
2
1
2
1
2



 m
L
L
 one of two reflected rays has phase-shift
 Destructive condition: 
m
L
L 
 1
2
 Incidence light to film surface is nearly
normal  2  0 & cos2  1  path
difference L2 – L1 = 2n2tcos2 = 2nt
= n
 Constructive condition:
 
2
1
2
1
2



 m
L
L
 Both reflected rays has phase-shift
 Destructive condition:

m
L
L 
 1
2

5. 5
Light
source
Reflection
mirror
Glass
plates
Optical
Microscope
 Air film between either two glass plates or one upper convex glass len with
large curvature radius (~ m) and one lower glass plate (usually n > nair).
 Thickness t gradually increased from 0 to 1 m with very small inclined
angle ( ~ several percentage of degree)
Light
source
Reflection
mirror
Wegde-like air gap Convex
glass len
Glass
plate
Optical Microscope
Newton fringes

 Air wedge
2. Thin film interference

6. Incident
ray
nG
R
O
R - t
1
2
nG
t
rm
Reflection
rays
 Bright fringe:  
2
1
2
2
1
2




 m
t
L
L
 Dark fringe: L2 – L1 = 2t = m
 
4
1
2


 m
tb
m
 Wedge’s thickness at bright fringes:
 Wedge’s thickness at dark fringes:
2

m
td
m 
 Fringe’s width:
α
λ
α
sin
t
t
i m
m
2
1


 
 Dark Newton fringe’s radius: mRλ
Rt
r m
m 
 2
 Bright Newton fringe’s radius:
1st bright fringe m = 0
Central dark fringe,
m = 0
1st dark fringe m = 1
Dark
Bright
Contact edge
of G1 & G2
 tm
tm+1
G1
G2
l

If index of refraction is
gradually increased 
no any phase shift for
both reflected rays 
interference conditions
are reversed.
5th dark fringe, m = 4
G1 & G2
tm
air
2. Thin film interference
 Air wedge
 
2
1
2
2
Rλ
m
Rt
r m
m 

