Electromagnetic waves: Interference

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Electromagnetic waves Interference

• Wednesday November 6, 2002

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Haidingers Bands Fringes of equal inclination
d
n1
n2
Beam splitter
P
?1
x
?
?1
f
Extended source
Focal plane
Dielectric slab
PI
P2
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Fizeau Fringes fringes of equal thickness

• Now imagine we arrange to keep cos ? constant
• We can do this if we keep ? small
• That is, view near normal incidence
• Focus eye near plane of film
• Fringes are localized near film since rays
  diverge from this region
• Now this is still two beam interference, but
  whether we have a maximum or minimum will depend
  on the value of t

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Fizeau Fringes fringes of equal thickness
where,
Then if film varies in thickness we will see
fringes as we move our eye. These are termed
Fizeau fringes.
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Fizeau Fringes
Beam splitter
Extended source
n
n2
n
x
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Wedge between two plates
1
2
glass
D
y
glass
air
L
Path difference 2y Phase difference ?
2ky - ? (phase change for 2, but not for 1)
Maxima 2y (m ½) ?o/n Minima 2y m?o/n
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Wedge between two plates
Maxima 2y (m ½) ?o/n Minima 2y m?o/n
D
y
air
Look at p and p 1 maxima yp1 yp ?o/2n ?
?x? where ?x distance between adjacent
maxima Now if diameter of object D Then L?
D And (D/L) ?x ?o/2n or D ?oL/2n ?x
L
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Wedge between two plates
Can be used to test the quality of surfaces
Fringes follow contour of constant y Thus a flat
bottom plate will give straight fringes,
otherwise ripples in the fringes will be seen.
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Newtons rings
Used to test spherical surfaces
Beam splitter
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Newtons rings
Maxima when, y (m1/2) ?o/2n Gives
rings, Rm2(m1/2)?oR/n
R- Rcos? y or, R2(R-y)2r2 ? R2(1-2y/R)
r2
?
i.e. r2 2yR
R
R
y
r
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Reflection from dielectric layer Antireflection
coatings

• Important in instruments such as cameras where
  reflections can give rise to spurious images
• Usually designed for particular wavelength in
  this region i.e. where film or eye are most
  sensitive

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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