8' Total Internal Reflection and the Evanescent Wave

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8. Total Internal Reflection and the Evanescent
Wave

• Phase shifts in reflection
• Total internal reflection and applications
• The mysterious evanescent wave
• Metals and a complex refractive index

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Phase Shift in Reflection (for Perpendicularly
Polarized Light)

• So there will be destructive interference between
  the incident
• and reflected beams just before the surface.
• Analogously, if ni gt nt (glass to air), r? gt 0,
  and there will be
• constructive interference.

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Phase Shift in Reflection (Parallel Polarized
Light)

• BUT THE WAY WE DEFINED THE FIELD, THIS MEANS
• DESTRUCTIVE INTERFERENCE WITH INCIDENT BEAM!
• Analogously, if ni gt nt (glass to air), r lt
  0, and we have
• constructive interference.
• Good that we get the same result as for r? its
  the same problem when qi 0!!!!!

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Phase shifts in reflection (air to glass)

• ni lt nt

180 phase shift for all angles
180 phase shift for angles below Brewster's
angle 0 for larger angles
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Phase shifts in reflection (glass to air)
0 phase shift below Brewster's angle 180 above
it Complex above critical angle
ni gt nt
0 phase shift below critical angle
Complex above critical angle
Relative phase shift (interesting for
polari- zation rotation effects)
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Total Internal Reflection occurs just as
thetransmitted beam grazes the surface.

• Note that the irradiance of the transmitted beam
  goes to zero as it
• grazes the surface.

Total internal reflection is 100 efficient.
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Applications of Total Internal Reflection

• Beam steerers

Beam steerers used to compress the path
inside binoculars
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Frustrated Total Internal Reflection

• By placing another surface in contact with a
  totally internally
• reflecting one, the TIR can be frustrated.

This effect provides evidence for evanescent
fields and is the basis for a variety of
spectroscopic techniques.
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The Evanescent Wave

• The evanescent wave is the "transmitted wave"
  when total internal reflection occurs. A very
  mystical quantity! So we'll do a mystical
  derivation

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The Evanescent Wave k-vector

• The evanescent wave k-vector must have x and y
  components
• Along surface ktx kt sin(qt)
• Perpendicular to it kty kt cos(qt)
• Using Snell's Law, sin(qt) (ni /nt) sin(qi), so
  ktx is meaningful.
• And again cos(qt) 1 sin2(qt)1/2 1
  (ni /nt)2 sin2(qi)1/2
• i b
• Neglecting the unphysical ib solution, we have
• Et(x,y,t) E0 expby exp i (kt (ni /nt)
  sin(qi)x wt
• The evanescent wave decays exponentially in
  transverse direction.

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Optical Properties of Metals

• A simple model of a metal is a gas of free
  electrons (the Drude model).
• These free electrons and their accompanying
  positive nuclei can
• undergo "plasma oscillations" at frequency, wp.
• where

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Reflection from metals

• At normal incidence in air
• Generalizing to complex refractive indices