Coherence, Incoherence, and Light Scattering
Coherence, Incoherence, and Light Scattering
Coherence, Incoherence, and Light Scattering
Scattering
Coherence vs. incoherence.
Re
If we plot the
Waves adding exactly
complex
in phase (coherent
amplitudes: constructive addition)
I = I1 + I 2 + cε Re { E1⋅ E2*} E
~1
and E
~2
are complex amplitudes.
% %
Im
If we write the amplitudes in Ei
%
Ei ∝ I i exp[− iθi ]
terms of their intensities, Ii, A
θ
and absolute phases, θ i, % i
Re
%% %% % %
I1, I2, … In are the irradiances of the Ei Ej* are cross terms, which have the
various beamlets. They’re all phase factors: exp[i(θ i-θ j)]. When
positive real numbers and they add. the θ ’s are random, they cancel
out!
All the
Itotal = I1 + I2 + … + In relative Im
phases
Re
I1+I2+…+IN
The intensities simply add!
Two 20W light bulbs yield 40W.
exp[i (θi − θ j )] exp[i(θ k − θ l )]
Light bulbs
Itotal = I1 + I2 + … + In
Laser
Etotal = E1 + E2 + … + En Itotal = I1 + I2 + … + In
Light Scattering
Molecule
When light encounters
matter, matter not only re-
emits light in the forward Light source
direction (leading to
absorption and refractive
index), but it also re-emits
light in all other directions.
r
E ( r , t ) ∝ ( E0 / r ) Re{exp[i (kr − ω t )]}
Far away,
spherical wave-
fronts are almost
flat…
Etotal = E1 + E2 + … + En
I1, I2, … In are the irradiances of Ei Ej* are cross terms, which have the
the various beamlets. They’re all phase factors: exp[i(θ i-θ j)]. When the
positive real numbers and add. θ ’s are not random, they don’t cancel
out!
Itotal = I1 + I2 + … + In
To understand scattering in a given
situation, we compute phase delays.
Wave-fronts
Because the phase is
constant along a
L1
wave-front, we
compute the phase L2
delay from one wave-
L3 Potential
front to another wave-front
potential wave-front. L4
φi = k Li
Scatterer
If the phase delay for all scattered waves is the same (modulo 2π ),
then the scattering is constructive and coherent. If it varies uniformly
from 0 to 2π , then it’s destructive and coherent.
If it’s random (perhaps due to random motion), then it’s incoherent.
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 there’s a direction for which
coherent constructive interference occurs.
Consider the
different phase
delays for
different paths.
Potential
wave front
a
Looking from any other direction, you’ll see no light at all due to
coherent destructive interference.
Incoherent scattering: reflection from a
rough surface
This is why rough surfaces look different from smooth surfaces and
mirrors.
Why can’t we see a light beam?
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.
Constructive
interference will
occur for a
transmitted beam if
Snell's Law is
obeyed.
On-axis vs. off-axis light scattering
Forward (on-axis) light Off-axis light scattering: scattered
scattering: scattered wavelets have random relative
wavelets have nonrandom phases in the direction of interest
(equal!) relative phases in due to the often random place-ment
the forward direction. of molecular scatterers.
There will still be scattering from the surfaces because the air nearby
is different and breaks the symmetry!
Scattering from particles is much
stronger than that from molecules.
They’re bigger, so they scatter more.
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 isn’t smooth, the scattering is incoherent.
If the surfaces are smooth,
then we use Snell’s Law
and angle-of-incidence-
equals-angle-of-reflection.
Air
Geometrical optics
refractive index.
Rayleigh Scattering
Rainbow
Large
m = -1 m=0 m =1
m=2
Diffracted white light
High pressure (over time) squeezes the air bubbles out, leaving
molecular scattering as the main source of scattering.
Sunsets involve longer path lengths
and hence more scattering.
Note the cool sunset.
Noon ray
Sunset ray
Atmosphere