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 4

4

1

or

os

oi

E

ELet is the incident amplitude,

is the scattered amplitude at a distance r from the scatterer.

V is the volume of the scatterer.

Example 4.1

Establish the dependence of the percentage of light scattered in Rayleigh scattering.

Vr

EE oios

1

Assume

r

EVK

r

EVE oioi

os

4

r

EVKE oi

os

r

KV Must be unitless, and

K must has units of ( Length )2

4

2

1,

1

os

oioios

I

r

EV

r

EVKE

The Transmission of Light Through Dense Media

Interference produces a redistribution of energy, out of the regions where it’s destructive into the regions where it’s

constructive.

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.

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

Scattering from an individual molecule or particle is weak, but many such scatterings can add up—especially if interference is coherent and constructive.

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).

A plane wave impinging on a molecule or particle scatters into a spherical wave.

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:

•

0 exp( ) E A i

(note the " ~ ")

0, expE x t E i kx t

As written, this entire field is complex!

Complex numbers simplify optics!

This isn't so obvious using trigonometric functions, but it's easywith complex exponentials:

1 2 3

1 2 3

( , ) exp ( ) exp ( ) exp ( )

( ) exp ( )totE x t E i kx t E i kx t E i kx t

E E E i kx t

where all initial phases are lumped into E1, E2, and E3.

Adding waves of the same frequency, but different initial phase, yields a wave of the same frequency.

When two waves add together with the same complex exponentials,

we add the complex amplitudes, E0 + E0'.

Adding complex amplitudes

Slower phase velocityLaser Absorption

+

=

time

1.0

-0.2

0.8

Destructive interference:

1.0

0.2

1.2

+

=

time

Constructive interference:

+

=

time

"Quadrature phase" ±90° interference:

1.0

-0.2i

1-0.2i

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:

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.

Electric field at atom

Electron cloud

Emitted field

On resonance (= 0)

( )ex t

( )E t

( )E t +

=

Incident light

Emitted light

Transmitted light

The interaction of light and matterLight excites atoms, which then emit more light.

The crucial issue is the relative phase of the incident light and this re-emitted 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.

Electric field at atom

Electron cloud

Emitted electric field

On resonance (the light frequency is the same as that of the atom)

( )ex t

( )E t

( )E t +

=

Incident light

Emitted light

Transmitted light

The relative phase of emitted light with respect to the input light depends on the frequency.

Below resonance

<< 0

Electric field at atom

Electron cloud

On resonance= 0

Above resonance

>> 0

The emitted light is 90° phase-shifted with respect to the atom’s motion.

Emitted field

Weak emission.90° out of phase.

Strong emission.180° out of phase.

Weak emission.-90° out of phase.

Refractive index and Absorption coefficient

2 20

2 2 2 20 0 0 0 0

/ 2 1

2 ( ) ( / 2) 4 ( ) ( / 2)e e

Ne Nen

c m m

0

Absorption coefficient

Refractive index

0

n–1

Frequency, Frequency, 0

Variation of the refractive index with wavelength (dispersion) causes the

beautiful prismatic effects we know and love.

Prisms disperse white light into its various colors.

Prism

Input white beam

Dispersed beam

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.

Light scattering regimes

Particle size/wavelength

Re

frac

tive

inde

x

Mie Scattering

Ra

yle

igh

Sca

tteri

ng

Totally reflecting objects

Ge

om

etr

ica

l opt

ics

Rayleigh-Gans Scattering

Larg

e

~1

~

0

~0 ~1 Large

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:

Rainbow

Air

Mie Scattering.

                                                                                                          

The mathematics of scattering

Itotal = I1 + I2 + … + In

I1, I2, … In are the irradiances of the various beamlets. They’re all positive real numbers and add.

* * *1 2 1 2 1 3 1... Re ...total N N NI I I I c E E E E E E

If the phases aren’t random, we add the fields:

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:

Coherent

Incoherent

Etotal = E1 + E2 + … + En