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Chapter 30
The Wave Nature of Light;
Interference
Huygens’ principle
Nature of light: particles or wave?
Huygens’ principle:
Points on wave front → sources of tiny wavelets
New wave front → envelope of all the wavelets
(tangent to all of them)
New wave front
& traveling direction
2
Wave Nature of Light
Diffraction:
Refraction:
Index of refraction:
n  c/v
3
Interference conditions
1) Same frequency (monochromatic)
2) Same oscillation direction (polarization)
3) Constant phase difference
Natural light source: continuous spectrum
Monochromatic: a very narrow range of λ
4
Coherence
Light sources produce “wave trains” of light
·
·
(random phase
relation)
independent
A great many of such wave trains → incoherent
Constant phase relation → coherent
1) laser
2) separate from one source
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Typical types of interference
Interference of two beams from one source:
P
S*
(1) Two points on the
same wave front
(double-slit)
S*
·P
(2) Light reflected by
two surfaces
(thin films)
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Interference of light
Coherent lights from same source in vacuum:
Amplitude:
A  A1  A2  2 A1 A2 cos  
2
2
2
where    k ( r2  r1 ) 
r

Intensity:
Δr: Path difference
I  I 1  I 2  2 I 1 I 2 cos  
1)    2m    r  m   I max
1
2)    (2 m  1)   r  ( m  )   I min
2
bright
dark
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Young’s double-slit experiment
Classical experiment in the history
Convincing evidence for wave nature of light
Equipment
& Results
Plane monochromatic light falls on the two slits
8
Graphs & intensity
Bright
Intensity
Dark
Diffraction
9
Interference patterns
Path difference
 r  r2  r1  d sin 
Bright lines
d sin   m 
m=0, 1, 2, … Order of the interference fringes
Center fringe is a bright line: d sin   0
1
Dark lines d sin   ( m  ) 
2
m=0, 1, 2, …
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Positions of fringes
For small angles
sin   tan   
Bright fringes:
x
d sin   d  m 
L
L
m=0, 1, 2, …
 x   m ,
d
Dark fringes: x  L ( m  1 )  , m=0, 1, 2, …
d
2
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Colorful fringes
Example1: White light(4000Å ~7000Å) passes two
slits 0.25mm apart and the fringes is shown on a
screen 100cm away. What is the width of the first
colorful band? (1Å=10-10m)
Solution: Why there is a spectrum?
Bright fringes: x  L  m  / d , m  1
For 4000Å: x1  L  1 / d  1.6mm
x
o
For 7000Å: x 2  L   2 / d  2.8mm
Width:
 x  x 2  x1  1.2mm
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Changing the conditions
Thinking: What happens to the interference pattern
if: (a) two slits moves further apart; (b) the screen
moves further apart; (c) a thin piece of glass is
placed in front of one slit.
Answer: bright fringes
x  L  m / d
So d ↗ causes x ↘, closer
L ↗ causes x ↗, further
How about put a glass?
d
L
Fringes will move upward
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Mediums and OPD
Speed of light in a medium is less than c
Index of refraction: n  c / v
Wavelength in medium:  n  v / f   / n
λ is the wavelength of light in vacuum
Phase difference:  
2
2
r 
 nr
n

nr: distance traveled in vacuum in same time
Optical path difference (OPD):
  nr
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Calculate OPD
Question: What is the optical path difference?
s1
n1
n2
s2
p
r1
p
r2
 = n1r1- n2r2
s1
S1 p= r1
s2
S2 p= r2
 = (r1-t1 +n1t1) - (r2-t2 +n2t2)
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Interference in thin films
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Reflections and interference
Two reflection lights
Notice: phase change of π
or half-wave loss
in reflection if n1< n2 !

n1
n2
n3
A
C
t
B
bright
m


2n2t (  2 )  
1
(m  ) dark


2
If : n1 > n2 < n3 or n1 < n2 > n3 , +/2
If : n1 > n2 > n3 or n1 < n2 < n3 , no change!
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Coating of glass

bright
m


(

)
2n2t 2  
1
(m  ) dark


2
Thin material deposited on the lens surface
Reflection decreases → nonreflective coatings
Reflection increases → reflective coatings
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Coating example
Example2: A film coating of MgF2 (n2=1.38) is on a
glass (n3=1.50). What is the thickness to eliminate
reflected light at wavelength centered at 550nm?
1 1 
  2n2t (  m)?  
2 2 
For a minimum thickness
Solution:

t
 99.6nm
4n2
What color does it look?
n1
n2
n3
A
C
air
MgF2
B
t
glass
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Color of a bubble
Example3: White light is incident on a soap bubble,
the thickness t=3800Å, n=1.33, find the color of
bubble in reflecting light.
Solution:
  2n2t  / 2  m
o
2 n2 t
7600 A1.33

 
1
1
m

m
2
2
For visible light (4000Å~7500Å)
m=1,…
m=2, =6739Å red
m=3, =4043Å purple
Color in different angle
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Wedge-shaped film
O

x
m  1, 2...
bright
m
 
  2t   
1
2 (m  ) m  0,1... dark

2
glass

t air h
t  tm 1  tm 
glass
2
l
Distance between fringes
t

L

sin  2sin 
It is a dark fringe on the edge
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Smoothness of workpiece
Example4: Check the level of smooth of workpieces
by using interference. Is it concave or protruding?
Determine the depth (or height) H.
l
Solution: Same fringe
→ Same thickness of air
a
H=asin
lsin = /2
a 
H

l 2
Standard level
H

workpiece concave
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Newton’s rings
m  1, 2...bright
m
 
  2t   

2  (2m  1)
m  0,1...dark

2
2
r
r 2  R2  ( R  t )2  2Rt  t 
2R


 (m  1 2) R , m  1, 2,... bright
r
mR ,
m  0,1,... dark


R
Convex surface
of a lens
Plane glass
plate
r
t
O
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Michelson Interferometer
Treated as an interference in thin film
  2( L2  L1 )  2t
bright
 m



movable
mirror
M2
(2m  1)
dark


2
L2
M 2 M 1
A
Precise measurement
Michelson-Morley
Experiment
C
P
D
fixed
mirror
Beam
splitter
L1
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*Interferometer & Images
equal inclination interference
Michelson Interferometer
equal thickness interference
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