Coherence, Incoherence, And Light Scattering

Coherence vs. incoherence.

Coherence in light sources.

Light bulbs vs. lasers.

Coherence in light scattering.

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.

Coherence, Incoherence, and Light Scattering

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

Interfering many waves: in phase, out of phase, or with random phase…

Waves adding exactly in phase (coherent constructive addition)

Waves adding with random phase, partially canceling (incoherent addition)

If we plot the complex amplitudes:

Re

Im

Waves adding exactly out of phase, adding to zero (coherent destructive addition)

The relative phases are the key.

*1 2 1 2ReI I I c E E

Recall that the irradiance of the sum of two waves is:

If we write the amplitudes in terms of their intensities, Ii, and absolute phases, i,

2 1 21 1 22 Re exp[ ( )]I I I I iI

exp[ ]i i iE I i

Imagine adding many such fields. In coherent interference, the i – j will all be known.

In incoherent interference, the i – j will all be random.

Re

Im

Ai

iE

Re

Im

1 – 2

1 22 I I

I0

1 2I I

E1 and E2 are complex amplitudes.~ ~

Adding many fields with random phases

Itotal = I1 + I2 + … + In

I1, I2, … In are the irradiances of the various beamlets. They’re all

positive real numbers and they add.

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

We find:

1 2[ ... ] exp[ ( )]total NE E E E i k r t

Ei Ej* are cross terms, which have the

phase factors: exp[i(i-j)]. When the ’s are random, they cancel out!

Re

ImAll the relative phases

exp[ ( )]i ji exp[ ( )]k li

The intensities simply add!Two 20W light bulbs yield 40W.

I1+I2+…+IN

Light bulbs

Light from a light bulb is very complicated!

1. It has many colors (it’s white), so we have to add waves of many different values of (and hence k-magnitudes).

2. It isn’t a point source, so, for each color, we have to add waves with many different k directions.

3. Even along one direction, many different molecules are emitting light with random relative phases (the effect we just considered).

Light from a light bulb is incoherent

Itotal = I1 + I2 + … + In

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.

Other characteristics of incoherent light:

1. It’s relatively weak.

2. It’s omni-directional.

3. Its irradiance is proportional to the number of emitters.

Coherent vs. Incoherent Light

Itotal = I1 + I2 + … + InEtotal = E1 + E2 + … + En

Coherent light:

1. It’s strong.

2. It’s uni-directional.

3. Total irradiance N2 or 0.

4. Total irradiance is the mag-square of the sum of individual fields.

Incoherent light:

1. It’s relatively weak.

2. It’s omni-directional.

3. Total irradiance N.

4. Total irradiance is the sum of individual irradiances.

Laser

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 source

Molecule

Spherical waves

where k is a scalar, and r is the radial magnitude.

Unlike a plane wave, whose amplitude remains constant as itpropagates, a spherical wave weakens. Its irradiance goes as 1/r2.

A spherical wave has spherical wave-fronts.

0( , ) / Re{exp[ ( )]}E r t E r i kr t

Note that k and r are not vectors here!

A spherical wave is also a solution to Maxwell's equations and is a good model for the light scattered by a molecule.

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.

We’ll check the interference one direction at a time, usually far away.

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.

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…

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

The math of light scattering is analogous to that of light sources.

To understand scattering in a given situation, we compute phase delays.

Because the phase is constant along a wave-front, we compute the phase delay from one wave-front to another potential wave-front.

If the phase delay for all scattered waves is the same (modulo 2), then the scattering is constructive and coherent. If it varies uniformly from 0 to 2, then it’s destructive and coherent.

If it’s random (perhaps due to random motion), then it’s incoherent.

i ik L

Wave-fronts

L4

L2

L3

L1

Scatterer

Potentialwave-front

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.

Coherent constructive interference occurs for a reflected beam if the angle of incidence = the angle of reflection: i = r.

i r

The wave-fronts are perpendicular to the k-vectors.

Consider the different phase

delays for different paths.

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.

Coherent destructive interference occurs for a reflected beam direction if the angle of incidence ≠ the angle of reflection: i ≠ r.

i too big

Potential wave front

a

= ka sin(i) = ka sin(too big)

Coherent scattering 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, you’ll see no light at all due to coherent destructive interference.

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 we’ll see weak light in all directions.

This is why rough surfaces look different from smooth surfaces and mirrors.

Why can’t we see a light beam?

To photograph light beams in laser labs, you need to blow some smoke into the beam…

Unless the light beam is propagating right into your eye or is scattered into it, you won’t 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 (don’t try this at home…)

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.

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 scattering is coherent—even if the scatterers are randomly arranged in space.

Path lengths are equal.

Off-axis scattering is incoherentwhen the scatterers are randomly arranged in space. Path lengths are random.

Forward (on-axis) light scattering: scattered wavelets have nonrandom (equal!) relative phases in the forward direction.

Scattering from a crystal vs. scattering from amorphous material (e.g., glass)

A perfect crystal has perfectly regularly spaced scatterers in space.

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!

So the scattering from inside the crystal cancels out perfectly in all directions (except for the forward and perhaps a few other preferred directions).

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.

Then we add up all the waves resulting from all the input waves, taking into account their coherence, too.

Light scattering regimes There are many regimes of particle scattering, depend-ing on the particle size, the light wave-length, and the refractive index. You can read an entire book on the subject:

Particle size/wavelength

Re

lativ

e r

efra

ctiv

e in

de

x

Mie Scattering

Ra

yle

igh

Sca

tteri

ng

Totally reflecting objectsG

eo

me

tric

al o

ptic

s

Rayleigh-Gans Scattering

Larg

e

~

1

~

0

~0 ~1 Large

Rainbow

Air

This plot considers only single scattering by spheres. Multiple scattering and scattering by non-spherical objects can get really complex!

Diffraction Gratings

Scattering ideas explain what happens when light impinges on a periodic array of grooves. Constructive interference occurs if the delay between adjacent beamlets is an integral number, m, of wavelengths.

sin( ) sin( )m ia m

where m is any integer.

A grating has solutions or zero, one, or many values of m, or orders.

Remember that m and m can be negative, too.

Path difference: AB – CD = m

Scatterer

Scatterer

a

i

m

a

AB = a sin(m)

CD = a sin(i)

A

DC

B

Potential diffracted wave-front

Incident wave-front

i

m

Diffraction orders

Because the diffraction angle depends on , different wavelengths are separated in the nonzero orders.

No wavelength dependence occurs in zero order.

The longer the wavelength, the larger its deflection in each nonzero order.

Diffraction angle, m()

Zeroth order

First order

Minus first order

Incidence angle, i

Because diffraction gratings are used to separate colors, it’s helpful to know the variation of the diffracted angle vs. wavelength.

Differentiating the grating equation, with respect to wavelength:

Diffraction-grating dispersion

cos( ) mm

da m

d

sin( ) sin( )m ia m

cos( )m

m

d m

d a

Rearranging:

[i is constant]

Thus, to separate different colors maximally, make a small, work in high order (make m large), and use a diffraction angle near 90 degrees.

Gratings typically have an order of magnitude more dispersion than prisms.

Any surface or medium with periodically varying or n is a diffraction grating.

Transmission gratings can be amplitude () or phase (n) gratings.

Gratings can work in reflection (r) or transmission (t).

Real diffraction gratings

Diffracted white light

White light diffracted by a real grating.

m = 0 m =1m = 2

m = -1

The dots on a CD are equally spaced (although some are missing, of course), so it acts like a diffraction grating.

Diffraction gratings

World’s largest diffraction grating

Lawrence Livermore National Lab

Wavelength-dependent incoherent molecular scat-tering: Why the sky is blue.

Air molecules scatter light, and the scattering is proportional to 4.

Shorter-wavelength light is scattered out of the beam, leaving longer-wavelength light behind, so the sun appears yellow.

In space, there’s no scattering, so the sun is white, and the sky is black.

Light from the sun

Air

Why is some ice blue?

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.

As you know, the sun and clouds can appear red.

Edvard Munch’s “The Scream” was also affected by the eruption

of Krakatoa, which poured ash into the sky worldwide.

Munch Museum/Munch Ellingsen Group/VBK, Vienna

Sunset ray

Atmosphere

Note the cool sunset.Noon ray