Application: Anti-Reflective Coatings

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Application Anti-Reflective Coatings

• Your eyeglasses (possibly) and sophisticated
  multi-element optical systems like telephoto
  lenses (definitely) rely on anti-reflective
  coatings to reduce extraneous images, loss of
  contrast, and other image degradation due to
  unwanted reflections.
• Straightforward application of thin-film
  interference

• Coating of SiO, MgF2 or other. Hard, transparent,
  and easy to apply in a thin uniform coating (e.g.
  by vapor deposition).
• nair lt ncoating lt nglass two phase jumps
• Thickness ?/4 for best transmission (destructive
  interference for reflection) where l is the
  wavelength in the film.
• Explicitly, lfilmlair/n
• Single-layer coating optimized for one
  wavelength, typically 550 nm (yellow/green).
  Visible color in white light is
  white-yellow/green ? purple

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How is the energy of an incident wave shared
between reflected and transmitted waves at a
boundary?

• Derivable from Maxwells equations and physical
  boundary conditions on the E and B fields.

Polarized along x. Propagating toward z.
Normally incident on boundary at z z0 between
region 1 (n1) and region 2 (n2).
Apply these to a harmonic EM wave crossing this
bdy.
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Example - Air(n11) Lucite(n21.5)
R2.04 T2.96
Analysis is similar, but more complicated, for
non-normal incidence ? Fresnel formulas.
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Michelson Interferometer

• Interferometer sensitive instrument that can
  measure physical parameters that change the phase
  of a wave.
• Path length (distance), refractive index, motion
  with respect to wave medium,
• Michelsons interferometer produces interference
  fringes by splitting a monochromatic beam,
  sending the two parts along different paths, and
  recombining to form an interference pattern. One
  path has a movable mirror (the other has a fixed
  one). Precise distance measurements are made by
  moving the mirror and counting the number m of
  interference fringes that pass a reference and
  determining the distance as d m?/2.

Demonstration Michelson Interferometer
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Closer Look
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Diffraction

• More of the same! Interference in different
  circumstances.
• News flash Light does not really propagate like
  a geometrical ray or simple particle.
• Demonstration Laser and Diffraction Objects
• In coherent, monochromatic, light, razor blades
  cast fuzzy shadows, pinholes produce interference
  fringes, and solid spheres appear to have holes.

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The Poisson Spot
The center of the shadow is the only point that
is equidistant from every point on the
circumference, so there must be constructive
interference.
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Diffraction around knife edges, through slits and
pinholes, and in other circumstance were studied
extensively by Fresnel, both experimentally and
by Huygens construction.
Fresnel Diffraction is the general case, with
full detailed analysis and diffraction patterns
that depend on distance from the obstacle.
Fraunhofer Diffraction refers to the simplified
case where the light source and observation point
are both far from the obstacle, so that plane
waves can be assumed and rays taken to be
parallel. This is the approximation we use.
(Essentially the small-angle approximation we
used before.)
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Multiple-Slit Interference
Two slits were nice, how about three?

• Assume coherent, monochromatic waves. Waves
  through S1 interfere constructively with those
  through S2 when
• d sin ? n ?(n 0, ?1, ?2,)
• Waves from S2 interfere constructively with those
  from S2 when
• d sin ? n ?(n 0, ?1, ?2,)
• The maxima overlap, but the minima are more
  complicated.
• When S1 and S2 cancel, S3 is left over. Full
  cancellation occurs when S1S2 cancels S3 or
  S2S3 cancels S1.
• Full cancellation occurs when there is a 120? (or
  240? or) phase spacing of the waves from the
  sources.

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Abbreviated derivation
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Full intensity analysis
Observed intensity pattern
Principal Maxima
Secondary Maxima
Demonstration Multiple Slits
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We didnt go through this exercise because we
care about 3 slits!N parallel slits makes a

• Diffraction Grating

• Principal maxima
• d sin ? m ?
• (m 0, ?1, ?2, is the order)
• Minima
• d sin ? m ?/N
• (m 0, ?1, ?2, as above, with m N, 2N, are
  excluded)

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• Diffraction Grating Intensity Distribution

• Intensity (from superposition of fields)
• Imax N2 I0
• ltIgt N I0
• Width of maxima
• ?? ?/(N d)
• (sharper for big N)

Principal Maxima
As N gets bigger, the maxima get sharper

• What does a diffraction grating do with white
  light?
• Each wavelength gets its own set of principal
  maxima at characteristic angles.
• White light is dispersed into its component
  wavelengths a spectrum like that from
  dispersion, only better.

Secondary maxima are negligible for a real
grating with N of thousands.
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Demonstration Projected Spectra with Grating

• Anyone have a diffraction grating?
• A CD/DVD is an inadvertent reflection grating
• A real diffraction grating is a plate of optical
  glass, ruled with thousands of parallel grooves,
  or a plastic cast of a ruled glass grating. With
  light passing through you have a transmission
  grating and with a coating of aluminum you get a
  reflection grating.