Transcript 6. Light Scattering, Reflection, and Refraction

The Propagation of Light
The processes of transmission, reflection, and
refraction are macroscopic manifestations of
scattering occurring on a submicroscopic level.
Elastic Scattering
• In elastic scattering, the energy of the incident photon is
conserved and its propagating direction is changed by the
potential of the target .
Rayleigh Scattering
When a photon penetrates into a medium
composed of particles whose sizes are much
smaller than the wavelength of the incident
photon, the scattering process is elastic and is
called Rayleigh scattering. In this scattering
process, the energy (and therefore the
wavelength) of the incident photon is
conserved and only its direction is changed.
In this case, the scattering intensity is
proportional to the fourth power of the
reciprocal wavelength of the incident photon.
The scattering of electromagnetic
radiation by particles with
dimensions much smaller than the
wavelength of the radiation,
resulting in angular separation of
colors and responsible for the
reddish color of sunset and the
blue of the sky.
The intensity of the scattered light

1
4
or   4
Example 4.1
Establish the
scattering.
Let
 4
dependence of the percentage of light scattered in Rayleigh
Eoi is the incident amplitude,
Eos is the scattered amplitude at a distance r from the scatterer.
V
Assume
is the volume of the scatterer.
Eos  Eoi
1

r
V
V Eoi
V Eoi
Eos 
K
r
r
V Eoi
Eos  K
r
VK

r
Must be unitless, and
K must has units of ( Length )2
 V Eoi   V Eoi  1
 Eos  K 

 2
 r   r 
1
, I os  4

The Transmission of Light Through
Dense Media
Little or no light ends up scattered laterally or backwards
in a dense homogeneous medium.
This makes sense from the perspective of conservation of energy– we
can’t have constructive interference in every direction.
Interference produces a redistribution of energy, out of the
regions where it’s destructive into the regions where it’s
constructive.
Constructive vs. destructive interference;
Coherent vs. incoherent interference
Waves that combine
in phase add up to
relatively high irradiance.
Waves that combine 180°
out of phase cancel out
and yield zero irradiance.
Waves that combine with
lots of different phases
nearly cancel out and
yield very low irradiance.
=
Constructive
interference
(coherent)
=
Destructive
interference
(coherent)
=
Incoherent
addition
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 there’s
a molecule or particle
involved).
Scattering from an individual molecule or particle is weak, but
many such scatterings can add up—especially if interference is
coherent and constructive.
The Phases of the wavelets
at P differ greatly
The Transmission of
Light Through Dense
Media
Waves using complex amplitudes
•
•
The resulting "complex amplitude" is:
E0  A exp(i )
 (note the " ~ ")
E  x, t   E0 exp i  kx   t 
As written, this entire field
is complex!
Complex numbers simplify optics!
Adding waves of the same frequency, but different initial phase,
yields a wave of the same frequency.
This isn't so obvious using trigonometric functions, but it's easy
with complex exponentials:
Etot ( x, t )  E1 exp i(kx   t )  E2 exp i(kx   t )  E3 exp i (kx   t )
 ( E1  E2  E3 ) exp i(kx   t )
where all initial phases are lumped into E1, E2, and E3.
Adding complex amplitudes
When two waves add together with the same complex exponentials,
we add the complex amplitudes, E0 + E0'.
Constructive
interference:
1.0
Destructive
interference:
1.0
1.0
+
+
0.2
-0.2
=
1.2
Laser
+
-0.2i
=
=
0.8
time
"Quadrature phase" ±90°
interference:
1-0.2i
time
Absorption
time
Slower phase velocity
Light excites atoms, which emit light that adds
(or subtracts) with the input light.
When light of frequency  excites an atom with resonant frequency 0:
Electric field
at atom
Electron
cloud
Emitted
field
Incident light
E (t )
xe (t )
E (t )
+
Emitted light
=
On resonance ( = 0)
Transmitted light
An excited atom vibrates at the frequency of the light that excited it
and re-emits the energy as light of that frequency.
The crucial issue is the relative phase of the incident light and this
re-emitted light. For example, if these two waves are ~180° out of
phase, the beam will be attenuated. We call this absorption.
The interaction of light and matter
Light excites atoms, which then emit more light.
Electric field
at atom
Electron
cloud
Emitted
electric field
Incident light
+
E (t )
xe (t )
E (t )
Emitted light
=
On resonance (the light frequency is the
same as that of the atom)
Transmitted light
The crucial issue is the relative phase of the incident light and this reemitted light. If these two waves are ~180° out of phase, destructive
interference occurs, and the beam will be attenuated—absorption.
If they’re ~±90° out of phase: the speed of light changes—refraction.
The relative
phase of emitted
light with respect
to the input light
depends on the
frequency.
Electric field Electron Emitted
at atom
cloud
field
Below
resonance
 << 0
Weak
emission.
90° out of
phase.
On resonance
Strong
emission.
180° out
of phase.
Above
resonance
Weak
emission.
-90° out
of phase.
 = 0
The emitted light is
90° phase-shifted
with respect to the
atom’s motion.
 >> 0
Refractive index and Absorption coefficient
0
Ne2
 /2

2 0c0 me (0   ) 2  ( / 2) 2
Frequency, 
0  
Ne2
n 1 
4 0 me (0   ) 2  ( / 2) 2
Variation of the refractive index with
wavelength (dispersion) causes the
beautiful prismatic effects we know and
love.
Input
white
beam
Dispersed beam
Prism
Prisms disperse white light into its various colors.
Light Scattering
When light encounters
matter, matter not only reemits 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.
Light scattering regimes
~0
~1
Rayleigh Scattering
~1
Large
Refractive index
Air
Large
Rayleigh-Gans Scattering
Geometrical optics
~0
Particle size/wavelength
Mie Scattering
Rainbow
Totally reflecting objects
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:
Mie Scattering
.
The mathematics of scattering
If the phases aren’t random, we add the fields:
Coherent
Etotal = E1 + E2 + … + En
I total  I1  I 2  ...  I N  c Re E1 E2*  E1 E3*  ...  EN 1EN* 
I1, I2, … In are the irradiances of
the various beamlets. They’re all
positive real numbers and add.
Ei Ej* are cross terms, which have the
phase factors: exp[i(i-j)]. When the ’s
are not random, they don’t cancel out!
If the phases are random, we add the irradiances:
Itotal = I1 + I2 + … + In
Incoherent