Coherence vs' incoherence'

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Coherence, Incoherence, and Light Scattering

• Coherence vs. incoherence.
• Spherical waves are solutions to Maxwells
  equations.
• Light bulbs
• Molecules scatter spherical waves
• Spherical waves can add up to plane waves.
• Reflected and diffracted beams at surfaces.
• Why the sky and swimming pools are blue.

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Constructive vs. destructive interferenceCoheren
t vs. incoherent interference
Constructive interference(coherent)
Waves that combine in phase add up to relatively
high irradiance.

Waves that combine 180 out of phase cancel out
and yield zero irradiance.
Destructive interference(coherent)

Waves that combine with lots of different phases
nearly cancel out and yield very low irradiance.
Incoherent addition

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Interfering many waves in phase, out of phase,
or with random phase
If we plot the complex amplitudes
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The relative phases will be the key.
Recall that the irradiance of the sum of two
waves is
If we write the fields in terms of their real
amplitudes, A, and absolute phases, qi,
Imagine adding many such fields. In coherent
constructive interference, the qi qj will all
be the same. In incoherent interference, the qi
qj will all be different.
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A spherical wave is also a solution toMaxwell's
equations.
Note that k and r are not vectors here!

• where k is a scalar, and
• r is the radial co-ordinate.

A spherical wave has spherical wave-fronts.
Unlike a plane wave, whose amplitude remains
constant as it propagates, a spherical wave
weakens. Its irradiance goes as 1/r2.
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Light bulbs
You might think that a spherical wave would be a
good model of a light bulb, which emits light in
all directions. But light from a light bulb is
much more complicated.
So we can model light from a light bulb using
plane waves, but its a very complicated sum of
them.
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Light from a light bulb
Just consider effect 3. Well have to add many
waves of the same real amplitude, k-vector, and
frequencybut with random phases.
I1, I2, In are the irradiances of the various
beamlets. Theyre all positive real numbers and
they add.
Ei Ej are cross terms, which have the phase
factors expi(qi-qj). When the qs are random,
they cancel out!
Possible relative phases

• Itotal I1 I2 In

The intensities simply add! Two 20W light bulbs
yield 40W.
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Light from a light bulb is incoherent
When many light waves add with random phases, we
say the light is incoherent, and the light wave
total irradiance is just the sum of the
individual irradiances.

• Itotal I1 I2 In

Other characteristics of incoherent light 1.
Its relatively weak. 2. Its omni-directional.
3. Its irradiance is proportional to the number
of emitters.
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Coherent vs. Incoherent Light

• Itotal I1 I2 In

Etotal E1 E2 En
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Light Scattering
When light encounters matter, matter not only
re-emits light in the forward direction (leading
to absorption and refractive index), but it also
re-emits light in all other directions.This is
called scattering.
Light scattering is everywhere. All molecules
scatter light. Surfaces scatter light. Scattering
causes milk and clouds to be white and water to
be blue. It is the basis of nearly all optical
phenomena.
Scattering can be coherent or incoherent.
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Light scattering regimes
There are many regimes of particle scattering,
depending on the particle size, the light
wavelength, and the refractive index. You can
read an entire book on the subject
Air
Rainbow
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The mathematics of scattering
If the phases arent random, we add the fields
Coherent
Etotal E1 E2 En
I1, I2, In are the irradiances of the various
beamlets. Theyre all positive real numbers and
add.
Ei Ej are cross terms, which have the phase
factors expi(qi-qj). When the qs are not
random, they dont cancel out!
If the phases are random, we add the irradiances
Incoherent

• Itotal I1 I2 In

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Scattering from molecules and small particles
A plane wave impinging on a molecule or particle
scatters into a spherical wave.
Huygens Principle says that waves propagate as
if each point on a wave-front emits a spherical
wave (whether or not theres a molecule or
particle involved).
Scattering from an individual molecule or
particle is weak, but many such scatterings can
add upespecially if interference is coherent and
constructive.
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Well check the interference one direction at a
time, usually far away.
This way we can approximate spherical waves by
plane waves in that direction, vastly simplifying
the math.
Far away, spherical wave-fronts are almost flat
Usually, coherent constructive interference will
occur in one direction, and destructive
interference will occur in all others. If
incoherent interference occurs, it is usually
omni-directional.
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To understand scattering in a given situation, we
compute phase delays.
Wave-fronts
Because the phase is constant along a wave-front,
we compute the phase delay from one wave-front to
another potential wave-front.
L1
L2
L3
Potentialwave-front
L4
Scatterer
If the phase delay for all scattered waves is the
same (modulo 2p), then the scattering is
constructive and coherent. If it varies
continuously from 0 to 2p, then its destructive
and coherent. If its random (perhaps due to
random motion), then its incoherent.
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Scattered spherical waves often combine to form
plane waves.
A plane wave impinging on a surface (that is,
lots of very small closely spaced scatterers!)
will produce a reflected plane wave because all
the spherical wavelets interfere constructively
along a flat surface.
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Coherent constructive scattering Reflection
from a smooth surface when angle of incidence
equals angle of reflection

• A beam can only remain a plane wave if theres a
  direction for which coherent constructive
  interference occurs.

Consider the different phase delays for different
paths.
Coherent constructive interference occurs for a
reflected beam if the angle of incidence the
angle of reflection qi qr.
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Coherent destructive scattering Reflection from
a smooth surface when the angle of incidence is
not the angle of reflection

• Imagine that the reflection angle is too big.
• The symmetry is now gone, and the phases are now
  all different.

qi
qtoo big
a
Coherent destructive interference occurs for a
reflected beam direction if the angle of
incidence ? the angle of reflection qi ? qr.
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Coherent scattering usually occurs in one (or a
few) directions, with coherent destructive
scattering occurring in all others.
A smooth surface scatters light coherently and
constructively only in the direction whose angle
of reflection equals the angle of incidence.
Looking from any other direction, youll see no
light at all due to coherent destructive
interference.
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Incoherent scattering reflection from a rough
surface
No matter which direction we look at it, each
scattered wave from a rough surface has a
different phase. So scattering is incoherent, and
well see weak light in all directions.
Coherent scattering typically occurs in only one
or a few directions incoherent scattering occurs
in all directions.
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Why cant we see a light beam?
Unless the light beam is propagating right into
your eye or is scattered into it, you wont see
it. This is true for laser light and
flashlights. This is due to the facts that air
is very sparse (N is relatively small), air is
also not a strong scatterer, and the scattering
is incoherent.
This eye sees almost no light.
This eye is blinded (dont try this at home)
To photograph light beams in laser labs, you need
to blow some smoke into the beam
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What about light that scatters on transmission
through a surface?

• Again, a beam can remain a plane wave if there is
  a direction for which constructive interference
  occurs.

Constructive interference will occur for a
transmitted beam if Snell's Law is obeyed.
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On-axis vs. off-axis light scattering

• Off-axis light scattering scattered wavelets
  have random relative phases in the direction of
  interest due to the often random place-ment of
  molecular scatterers.

Forward (on-axis) light scattering scattered
wavelets have nonrandom (equal!) relative phases
in the forward direction.
Forward scattering is coherent even if the
scatterers are randomly arranged in space.
Path lengths are equal.
Off-axis scattering is incoherent when the
scatterers are randomly arranged in space.
Path lengths are random.
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Scattering from a crystal vs. scattering from
amorphous material (e.g., glass)
A perfect crystal has perfectly regularly spaced
scatterers in space.
So the scattering from inside the crystal cancels
out perfectly in all directions (except for the
forward and perhaps a few other preferred
directions).
Of course, no crystal is perfect, so there is
still some scattering, but usually less than in a
material with random structure, like
glass. There will still be scattering from the
surfaces because the air nearby is different and
breaks the symmetry!
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Scattering from large particles
For large particles, we must first consider the
fine-scale scattering from the surface
microstructure and then integrate over the larger
scale structure. If the surface isnt smooth,
the scattering is incoherent. If the surfaces are
smooth, then we use Snells Law and
angle-of-incidence-equals-angle-of-reflection.
Then we add up all the waves resulting from all
the input waves, taking into account their
coherence, too.
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Diffraction Gratings

• If light impinges on a periodic array of grooves,
  scattering ideas tell us what happens. There
  will be constructive interference if the delay
  between adjacent beamlets is an integral number
  of wavelengths.

where m is any integer. A grating can have
solutions for zero, one, or many values of m, or
orders. Remember that m and the dif-fracted
angle can be negative, too.
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Diffraction orders
Because the diffraction angle depends on l,
different wavelengths are separated in the 1
(and -1) orders.
Diffraction angle, qm
First order
Zeroth order
Minus first order
No wavelength dependence in zero order.
The longer the wavelength, the larger its
diffraction angle in nonzero orders.
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Diffraction-grating dispersion
Its helpful to know the variation of the
diffracted angle vs. wavelength. Differentiating
the grating equation,
with respect to wavelength
qi is a constant
Rearranging
Gratings typically have an order of magnitude
more dispersion than prisms.
Thus, to separate different colors maximally,
make a small, work in high order (make m large),
and use a diffraction angle near 90 degrees.
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Wavelength-dependent incoherent molecular
scat-tering Why the sky is blue

• Air molecules scatter light, and the scattering
  is proportional to w4.

Shorter-wavelength light is scattered out of the
beam, leaving longer-wavelength light behind, so
the sun appears yellow. In space, the sun is
white, and the sky is black.
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Sunsets involve longer path lengths and hence
more scattering.
Note the cool sunset.
Noon ray
Sunset ray
Earth
Atmosphere
As you know, the sun and clouds can appear red.
Edvard Munchs The Scream was also affected by
the eruption of Krakatoa, which poured ash into
the sky worldwide.
Munch Museum/Munch Ellingsen Group/VBK, Vienna