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Chapter 1
INTERFERENCE OF LIGHT WAVES
OBJECTIVES
• To understand the principles of interference.
• To explain the intensity distribution in interference under
various conditions.
• To explain the interference from thin films.
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Wave optics (Physical Optics): The study of interference, diffraction, and
polarization of light. These phenomena cannot be adequately explained with the ray
optics.
Young’s Double-Slit Experiment
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To observe interference of waves from two sources, the following conditions must be
met:
• The sources must be coherent; that is, they must maintain a constant phase with
respect to each other.
• The sources should be monochromatic; that is, they should be of a single
wavelength.
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Analysis Model: Waves in Interference
Path difference
sin
1
2 d
r
r
Condition for constructive interference, at point P is,
...
,
2
,
1
,
0
sin bright
m
m
d
The number m is called the order number.
Condition for destructive interference, at point P is,
...
,
2
,
1
,
0
2
1
sin dark
m
m
d
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Linear positions of bright and dark fringes:
From the triangle OPQ,
L
y
tan
bright
bright tan
L
y
angles)
(small
bright
d
m
L
y
dark
dark tan
L
y
angles)
(small
2
1
bright
d
m
L
y
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Intensity Distribution of the Double-Slit Interference Pattern
Consider two coherent sources of sinusoidal waves such that they have same angular
frequency and phase difference .
t
E
E
t
E
E sin
and
sin 0
2
0
1
sin
2
2
d
t
t
E
E
E
EP sin
sin
0
2
1
2
sin
2
cos
2 0
t
E
EP
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Intensity of a wave is proportional to the square of the resultant electric field
magnitude at that point.
2 2 2 2
0
4 cos sin
2 2
P
I E E t
Most light-detecting instruments measure time-averaged light intensity, and the time
averaged value of
2
sin 2
t over one cycle is ½.
2
cos2
max
I
I
sin
cos2
max
d
I
I
L
y
sin
2
max cos
d
I I y
L
Alternatively, since
for small values of , we can write;
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Change of Phase Due to Reflection
Lloyd’s mirror. The reflected ray undergoes a
phase change of 180°.
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Interference in Thin Films
Consider a film of uniform thickness t and index of refraction n. Assume light rays
traveling in air are nearly normal to the two surfaces of the film. If is the wavelength
of the light in free space and n is the index of refraction of the film material, then the
wavelength of light in the film is
n
n
...)
,
2
,
1
,
0
(
2
1
2
m
m
t n
...)
,
2
,
1
,
0
(
2
1
2
m
m
nt
The condition for constructive interference
in thin films is,
The condition for destructive interference
in thin films is,
...)
,
2
,
1
,
0
(
2
m
m
nt
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Newton’s Rings
Expressions for radii of the dark rings:
Consider the dark rings (destructive interference)
...
3
,
2
,
1
,
0
,
2
m
m
nt
1
n
m
t
2
For air film,
From the above figure,
2
2
r
R
R
t
2
1
2
1
R
r
R
R
t
.......
!
2
)
1
(
1
1 2
y
n
n
ny
y
n
Binomial theorem is,
If / 1,
r R
R
r
R
r
R
R
t
2
........
2
1
1
2
2
...)
,
2
,
1
,
0
(
m
mR
rdark
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Michelson Interferometer
The interferometer, invented by A. A. Michelson,
splits a light beam into two parts and then
recombines the parts to form an interference pattern.
The interference condition for the two rays is
determined by the difference in their path length.
When the two mirrors are exactly perpendicular to
each other, the interference pattern is a target pattern
of bright and dark circular fringes.
If a dark circle appears at the center of the target
pattern and M1 is then moved a distance /4 toward
M0, the path difference changes by /2. This replaces
dark circle at center by bright circle. Therefore, the
fringe pattern shifts by one-half fringe each time M1 is
moved a distance /4.
