More Interference

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More Interference
Michelson Interferometer (cont'd) Unbalanced Ot
her interferometers Mach-Zehnder Sagnac Fizeau
Wedge Newton's Rings The Fabry-Perot
Interferometer Anti-Reflection
Coatings Single- and Multi-layer Photonic
Crystals
Prof. Rick Trebino Georgia Tech www.physics.gatech
.edu/frog/lectures
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The UnbalancedMichelson Interferometer
Misalign mirrors, so beams cross at an angle.

• Now, suppose an object isplaced in one arm.
  Now, one beam will have an extra
  spatiallyvarying phase, exp2if(x,y).
• The cross term becomes
• Re exp2if(x,y) exp-2ikx sinq

Place an object in this path
expif(x,y)
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The Unbalanced Michelson Interferometercan
sensitively measure phase vs. position.
Placing an object in one arm of a misaligned
Michelson interferometer will distort the spatial
fringes.

• Phase variations of a small fraction of a
  wavelength can be measured.

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The Mach-Zehnder Interferometer
The Mach-Zehnder interferometer is usually
operated misaligned and with something of
interest in one arm.
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Mach-Zehnder Interferogram
Nothing in either path
Plasma in one path
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The Sagnac Interferometer

• The two beams take the same path around the
  interferometer and the output light can either
  exit or return to the source.

Not only do both beams take the same path through
the device, they also pass through the same
amount of glass in the beam splitter!
It turns out that no light exits! It all returns
to the source!
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Why all the light returns to the source in a
Sagnac interferometer
Since the paths are the same, all that matters
are the phase shifts on the reflections. For the
exit beam Clockwise path has phase shifts of p,
p, p, and 0. Counterclockwise path has phase
shifts of 0, p, p, and 0. Perfect
cancellation!!
For the return beam Clockwise path has phase
shifts of p, p, p, and 0. Counterclockwise path
has phase shifts of 0, p, p, and p. Constructive
interference!
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The Sagnac Interferometer senses rotation.

• Suppose that the beam splitter moves by a
  distance, d, in the time, T, it takes light to
  circumnavigate the Sagnac interferometer.
• So one beam will travel more (d), and the other
  less distance (-d).
• If R the interferometer radius,
• and W its angular velocity
• Thus, the Sagnac Interferometer's sensitivity to
  rotation depends on its area. And it need not be
  round!

Sagnac Interfer- ometer (fiber)
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Newton's Rings
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Newton's Rings

• Get constructive interference when an integral
  number of half wavelengths occur between the two
  surfaces (that is, when an integral number of
  full wavelengths occur between the path of the
  transmitted beam and the twice reflected beam).

You see the color l when constructive
interference occurs.
You only see bold colors when m 1 (possibly 2).
Otherwise the variation with l is too fast for
the eye to resolve.
L
This effect also causes the colors in bubbles and
oil films on puddles.
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Radio-wave Interference in Audio Settings
Radio and acoustic waves can do the same.
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The Fizeau Wedge Interferometer
The Fizeau wedge yields a complex pattern of
variable-width fringes, but it can be used to
measure the wavelength of a laser beam.
Partiallyreflecting surface
Keep in mind that the input beam is large, so all
reflected beams interfere with each other.
Highly reflecting surface
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Multiple-beam interference The Fabry-Perot
Interferometer or Etalon

• A Fabry-Perot interferometer is a pair of
  parallel reflective surfaces. An etalon is a type
  of Fabry-Perot interferometer, and is a piece of
  glass with parallel sides.
• The transmitted wave is an infinite series of
  multiply reflected spatially overlapping beams.

r, t reflection, transmission coefficients from
glass to air and r, t reflection,
transmission coefficients from air to glass.
Transmitted wave E0t
Incident wave E0
Reflected wave E0r
n
nair 1
nair 1
d round-trip phase delay kL, where L is the
total effective round-trip path length.
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Lemma
Recall that
And the same for the perpendicular polarization
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Another result well need
Proof
Dividing both sides by (1-x)
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The Etalon (cont'd)

• The transmitted wave field is

tt 1 R
The transmittance is
Dividing numerator and denominator by
where
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Etalon Transmittance vs. Thickness, Wavelength,
or Angle
Transmission maxima occur when d / 2 mp pL/l
mp or

• The transmittance varies significantly with
  thickness, angle, and wavelength.
• As the reflectance of each surface (R) approaches
  1 (F increases), the widths of the
  high-transmission regions become very narrow.

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Does this look familiar?

• Recall that a finite train of identical pulses
  can be written
• where g(t) is a Gaussian envelope over the pulse
  train.

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The Etalon Free Spectral Range
The Free Spectral Range is the frequency or
wavelength range between transmission maxima.
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The Etalon Free Spectral Range
Alternative derivation (the more common one).
lFSR Free Spectral Range
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Etalon Line Width

• The line width dLW is a transmittance peak's
  full-width-half-max (FWHM).
• Setting d equal to dLW/2 should yield T 1/2

Transmittance
For d ltlt 1, we can make the small argument
approx
Substituting and we
have Or
The line width is the etalons wavelength-measurem
ent accuracy.
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The Interferometer or Etalon Finesse
The Finesse, F , is the ratio of the free
spectral range and the line width
Taking
The Finesse is the number of wavelengths the
interferometer can resolve.
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Focusing light into an etalon
If we focus light into an etalon, different
angles correspond to different path lengths. So
different colors will experience constructive
interference at different angles.
Focused white-light beam
Colorful rings!
Etalon
This image shows only one FSR, but typically many
will be evident.
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How to use an interferometer to measure wavelength
1. Measure the wavelength to within one Free
Spectral Range using a grating or prism
spectrometer to avoid the interferometers
inherent ambiguities. 2. Scan the spacing of the
two mirrors and record the spacing when a
transmission maximum occurs. 3. If greater
accuracy is required, use another (longer)
interferometer with a FSR the above accuracy
(line-width) and with an even smaller line-width
(i.e., better accuracy).
Interferometers are the most accurate measures of
wavelength available.
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Other applications of Fabry-Perot interferometers
and etalons
To frequency filter a beam (this is often done
inside a laser). Money is now coated with
interferometric inks to help foil counterfeiters.
Notice the shade of the 20, which is shown
from two different angles.
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Anti-reflection Coatings
Notice that the center of the round glass plate
looks like its missing. Its not! Theres an
anti-reflection coating there (on both the front
and back of the glass). Such coatings have been
common on photography lenses and are now common
on eyeglasses. Even my new watch is AR-coated!
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Anti-reflection Coatings

• Consider a beam incident on a glass substrate (n
  ns) with a layer of material (n nl) of
  thickness, h, on its surface.
• It can be shown that the reflectance is (for such
  thin media, we need to go back to Maxwells
  equations)

0 if
So an AR-coating requires
and
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Multilayer coatings

• Typical laser mirrors and camera lenses use many
  l/4 layers, usually with alternating high and low
  refractive indices.

The reflectance and transmittance vs. wavelength
can be tailored to taste!
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Stellar interferometry
Stars are too small to resolve using normal
telescopes, but interferometry can see them.
Stellar interferometers operate in the radio
(when the signals are combined electronically)
and visible (where the beams are combined
optically).
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Photonic crystals use interference to guide
lightsometimes around corners!
Yellow indicates peak field regions.
Borel, et al., Opt. Expr. 12, 1996 (2004)
Augustin, et al., Opt. Expr., 11, 3284, 2003.
Interference controls the path of light.
Constructive interference occurs along the
desired path.