Electromagnetic%20waves:%20Multiple%20beam%20Interference

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Electromagnetic waves Multiple beam Interference

• Wed. Nov. 13, 2002

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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
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,

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Multiple beam interference

• Thus maxima are circles in focal plane of lens
  or rings
• The maximum intensity
• And intensity distribution is,

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Multiple beam interference

• Minima intensity when,
• Intensity distribution,
• If R gt 0.9 IminltltImax

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Multiple beam interference

• Let the contrast, or co-efficient of finesse be
  defined by,
• Then the transmitted light is described by and
  Airy function,
• The same analysis for the reflected light gives

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Multiple beam interference Transmission curves
R0.046
F0.2
R0.18
F1
R0.87
F200
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Multiple beam interference Reflectivity curves
R0.87
F200
F1
R0.18
R0.046
F0.2
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Observation of fringe patterns
Screen or eye
source
f2
f1
Bright circles
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Typical parameters in an experiment

• Consider a quartz slab (n21.5) of thickness d
  0.5 cm
• The condition for constructive interference
  requires,
• For light of wavelength ?o 500 nm, incident at
  a small angle ?, i.e. ? also small, and m is
  large

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Spectroscopy applications Fabry- Perot
Interferometer

• Assume we have a monochromatic light source and
  we obtain a fringe pattern in the focal plane of
  a lens
• Now plot IT along any radial direction
• Let IMAXIM
• The fringes have a finite width as we scan

order
m
m-1
m-2
m-3
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Fabry-Perot Interferometer

• Full width at half maximum FWHM, is defined as
  the width of the fringe at I(½)IM
• Now we need to specify units for our application
• Let us first find ? such that I ½ IM

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Fabry-Perot Interferometer

• We want,
• This gives,

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Fabry-Perot Interferometer
m
m-1
I ½ IM
? 2(m-1)?
? 2m?
? 2(m-?m)?
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Fabry Perot Interferometer

• Thus at I ½ IM
• sin(?/2) sin (m ?m)? ? ? sin ?m?
• Assume ?m is small and
• sin ?m? ? ?m?
• Thus
• FWHM Fraction of an order occupied by fringe

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Fabry-Perot Interferometer

• The inverse of the FWHM is a measure of the
  quality of the instrument. This quality index is
  called the finesse
• It is the ratio of the separation between the
  fringes to the fringe width

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Fabry-Perot Interferometer

• Not that ? is determined by the reflectivity
• If R 0.90 ? 30
• R 0.95 ? 60
• R 0.97 ? 100
• In practive, cant get much better than 100 since
  the reflectivity is limited by the flatness of
  the plates (and other factors of course)

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Fabry-Perot Interferometer

• Now consider the case of two wavelengths (?1, ?2)
  present in the beam
• Assume ?1? ?2 and ?1lt ?2
• Increase ?2, dashed lines shrink
• e.g. order m-1 of ?2 moves toward mth order of ?1
• Eventually (m-1) ?2m ?1
• This defines the free spectral range

m
m-2
m-1
?2
?1
?2
?1
?2
?1
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Fabry-Perot Interferometer

• m(?2-?1) ?2 or m?? ?
• ??FSR ?/m
• Now since
• We have,
• e.g. ?500 nm, d 5mm, n1
• ? ??FSR 25(10-2)mm 0.25Å