Electromagnetic waves: Multiple beam Interference

1
Electromagnetic waves Multiple beam Interference

• Friday November 8, 2002

2
Anti-Reflection coatings
2
1
A. Determine thickness of film
air
n1
n2
film
n1 lt n2 lt n3
n3
glass
Thus both rays (1 and 2) are shifted in phase by
? on reflection.
For destructive interference (near normal
incidence) 2n2t(m1/2)? Determines the
thickness of the film (usually use m0 for
minimum t)
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Anti-Reflection coatings
2
1
air
B. Determine refractive index of film
n1
A
A
Near normal incidence Amplitude at A
n2
film
n3
glass
Since ? ? 1
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Anti-reflection coating
B. Determine refractive index of film
Amplitude at A
To get perfect cancellation, we would like EA E
A
should be index of AR film
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Multiple Beam interference

• Thus far in looking at reflectivity from a
  dielectric layer we have assumed that the
  reflectivity is small
• The problem then reduces to two beam interference
• Now consider a dielectric layer of uniform
  thickness d and assume that the reflectivity is
  large e.g. ? gt 0.8
• This is usually obtained by coating the surface
  of the layer with a thin metallic coating or
  several dielectric coatings to give high
  reflectivity
• Or, one can put coatings on glass plates , then
  consider space between plates

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Multiple beam interference
Let ?12 ? ?21 ? ?12 ? ?21 ?
??? Eo
(?)5??Eo
?Eo
(?)3??Eo
(?)7??Eo
?
n1
n2
?
A
B
C
D
n1
?
(?)2??Eo
(?)6??Eo
?? Eo
(?)4??Eo
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Multiple Beam Interference

• Assume a (for the time being) a monochromatic
  source
• ?, ? small ( lt 30o) usually
• Now ? ? gtgt ?, ?
• Thus reflected beams decrease rapidly in
  amplitude (from first to second)
• But amplitude of adjacent transmitted beam is
  about the same amplitude
• Amplitude of successfully reflected beams
  decreases slowly (from the second)
• Thus treat in transmission where contrast should
  be somewhat higher
• The latter is the configuration of most
  applications

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Multiple Beam Interference

• Assume phase of transmitted beam at A is such
  that,
• Now let ?be the phase shift in travelling across
  and back once, i.e.

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Multiple Beam interference

• At B
• At C
• At D

etc
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Multiple beam interference

• Consider N beams which interfere at infinity
• We can use a lens and then beams shown interfere
  in focal plane of lens
• The phase difference between adjacent rays
  outside is,

n1
n2
d
n1
?S1
N-4
N
N-3
N-1
N-2
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Multiple beam interference

• Amplitude at point P,

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Multiple beam interference

• Amplitude at point P,

Let
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Multiple beam interference
This is just a geometric series with r lt 1. Thus,
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Multiple beam interference
Thus,
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Multiple beam interference
Evaluate
Thus,
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Multiple beam interference
Now,
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Multiple beam interference
Now recall the definition of the intensity of an
electromagnetic wave
Thus,
is the intensity distribution in the focal plane
of the lens.
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Multiple beam interference
Fringe pattern
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Multiple beam interference

• Maximum intensity when,