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Purdue University
School of Mechanical Engineering
ME 587Engineering Optics
Lecture 16: Interference
ME 587Engineering Optics
Lecture 16: Interference
Prof. Robert P. Lucht
Room 86, Mechanical Engineering BuildingSchool of Mechanical Engineering
Purdue UniversityWest Lafayette, Indiana
[email protected], 765-494-5623 (Phone)
October 25, 2005
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Optical InterferenceOptical Interference
This Lecture• Two-Beam Interference• Young’s Double Slit Experiment• Virtual Sources• Newton’s Rings• Film Thickness Measurement by Interference
Last Lecture• Conduction Current in Metals• EM Wave Propagation in Metals• Skin Depth• Plasma Frequency
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Two-Beam InterferenceTwo-Beam Interference
1 2
1 01 1 1
2 02 2 2
:
, cos
, cos
Consider two waves E and E that have the same frequency
E r t E k r t
E r t E k r t
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Two-Beam InterferenceTwo-Beam Interference
1 2
1 2
2
20 0 1 2 1 2
2 20 1 2 1 2
1 2 12
:
, , ,
/ :
2
p
p
p
At a given position r the wave E is the superposition of E and E
E r t E r t E r t
The irradiance W m is given by
I c E c E E E E
c E E E E
I I I I
Hecht, Optics, Chapter 9.
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Two-Beam InterferenceTwo-Beam Interference
2 21 0 1 2 0 2
12 0 1 2
1 2 01 02 1 1 2 2
1 2 01 02 1
1 2 :
2
cos cos
cos cos cos sin sin
cos
The irradiances for beams and are given by
I c E I c E
The interference term is given by
I c E E
E E E E k r t k r t
But A B A B A B
E E E E k
1 1 1
2 2 2 2
2 201 02 1 1 2 2 1 1 2 2
1 1 2 2 1 1 2
cos sin sin
cos cos sin sin
cos cos cos sin sin sin
cos sin cos sin sin cos
r t k r t
k r t k r t
E E k r k r t k r k r t
k r k r t t k r k
2 cos sinr t t
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Two-Beam InterferenceTwo-Beam Interference
2 2
2 212 01 02 1 1 2 2 1 1 2 2
1 1 2 2
:
1 1cos sin cos sin 0
2 2
2 cos cos cos sin sin sin
cos sin cos sin sin
The time averages are given by
t t t t
The interference term is given by
I E E k r k r t k r k r t
k r k r t t
1 1 2 2
01 02 1 1 2 2 1 1 2 2
01 02 1 1 2 2
12 01 02 1 1 2 2
01 02
1
cos cos sin
cos cos sin sin
cos
cos
1
2
k r k r t t
E E k r k r k r k r
E E k r k r
I E E k r k r
If E and E are parallel then
I
2 20 01 2 0 02 12 1 2
1cos
2c E I c E I I I
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Two-Beam InterferenceTwo-Beam Interference
1 2 1 2
max 1 2 1 2 1 2 1 2
:
2 cos
0, 2 , :
2 cos 2 2
.
The total irradiance is given by
I I I I I
There is a maximum in the interference pattern when
I I I I I n I I I I
This is referred to as constructive interference
There is a minimum in the inter
min 1 2 1 2 1 2 1 2
1 2 0
20 0 0 max 0 min
, 3 , :
2 cos 2 1 2
.
2 2 cos 4 cos 4 02
ference pattern when
I I I I I n I I I I
This is referred to as destructive interference
When I I I
I I I I I I I
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Conditions for InterferenceConditions for Interference
Interference Between Light From Two Sources• Sources must be in phase with each other – sources must be coherent• Sources must have the same frequency• Radiation must overlap within the coherence lengths of each source to produce an interference pattern – discharge lamps have coherence lengths on the order of a few mm, some lasers have coherence lengths on the order of a few km• Clearest patterns will be produced when the sources have nearly the same amplitude – maximum contrast
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Conditions for InterferenceConditions for Interference
Hecht, Optics, Chapter 9.
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Young’s Double Slit Interference ExptYoung’s Double Slit Interference Expt
Hecht, Optics, Chapter 9.
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Young’s Double Slit Interference ExptYoung’s Double Slit Interference Expt
2 1
2 1
. :
sin
:
1sin
2
:
Assume that y s and a s The condition for an interference maximum is
S P S P m a
The condition for an interference minimum is
S P S P m a
Relation between
geometric path difference
and phase difference
2
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Young’s Double Slit Interference ExptYoung’s Double Slit Interference Expt
2 2 20 0 0
20
max
min
sin4 cos 4 cos 4 cos
2
:
sin tan 4 cos
: 0, 1, 2,
12:
On the screen the irradiance pattern is given by
aI I I I
Assuming that y s
y a yI I
s s
m sBright fringes y m
a
mDark fringes y
0, 1, 2,s
ma
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Interference Fringes From 2 Point SourcesInterference Fringes From 2 Point Sources
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Interference Fringes From
2 Point Sources
Interference Fringes From
2 Point Sources
Hecht, Optics, Chapter 9.
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Interference With Virtual Sources:Fresnel’s Double Mirror
Interference With Virtual Sources:Fresnel’s Double Mirror
Hecht, Optics, Chapter 9.
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Interference With Virtual Sources:Fresnel’s Biprism
Interference With Virtual Sources:Fresnel’s Biprism
Hecht, Optics, Chapter 9.
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Interference With Virtual Sources:Lloyd’s Mirror
Interference With Virtual Sources:Lloyd’s Mirror
Hecht, Optics, Chapter 9.
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Interference in Dielectric FilmsInterference in Dielectric Films
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Analysis of Interference in Dielectric FilmsAnalysis of Interference in Dielectric Films
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Analysis of Interference in Dielectric FilmsAnalysis of Interference in Dielectric Films
0
0
0 sin sin :
sin sin2
sin
sin2 sin
sin
p f
f f
i f t
t t
i
tt
The phase difference due to optical path length
differences for the front and back
reflections is given by
n AB BC n AD
n AE FC n AD n EB BF
But n n
ACAE AG
AD AC
AE AC AD
0
0 2
2
cos 2 cos
i f
f f
p f f
t p f t
nAD
n
n AD n AE n AE FC
n EB BF n EB
EB t n t
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Analysis of Interference in Dielectric FilmsAnalysis of Interference in Dielectric Films
:
2 cos
:
0, 1, 2,
:
1
2
r
f t r
p r
p r
Also need to account for phase differences
due to differences in the reflection process
at the front and back surfaces
n t
Constructive interference
m m
Destructive interference
m
0, 1, 2,m
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Fringes of Equal InclinationFringes of Equal Inclination
2 cos
:
0, 1, 2,
:
1
2
0, 1, 2,
f t r
p r
p r
Fringes arise as Δ varies due to changes
in the incident angle:
n t
Constructive interference
m
m
Destructive interference
m
m
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Fringes of Equal ThicknessFringes of Equal Thickness
2 cos
:
0, 1, 2,
:
1
2
f t r
p r
p r
When the direction of the incoming light is
fixed, fringes arise as Δ varies due to changes
in the dielectric film thickness :
n t
Constructive interference
m m
Destructive interference
m
0, 1, 2,m
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Fringes of Equal Thickness: Newton’s RingsFringes of Equal Thickness: Newton’s Rings
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Fringes of Equal Thickness: Newton’s RingsFringes of Equal Thickness: Newton’s Rings